The Interior-Point Revolution in Optimization: History, Recent Developments, and Lasting Consequences
Margaret H. Wright Computer Science Department Courant Institute of Mathematical Sciences New York University
AMS Special Session on Current Events Joint Mathematics Meeting January 9, 2004
Abstract Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interior-point techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded, nearly to the point of oblivion, by newly emerging and seemingly more effi- cient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization. This picture changed dramatically in 1984, when Narendra Karmarkar announced a fast polynomial-time interior method for linear programming; in 1985, a formal connec- tion was established between his method and classical barrier methods. Since then, in- terior methods have continued to transform both the theory and practice of constrained optimization. We present a condensed, unavoidably incomplete look at classical mate- rial and recent research about interior methods.
1 Overview
REVOLUTION: (i) a sudden, radical, or complete change; (ii) a fundamental change in political organization, especially the overthrow or renunciation of one government or ruler and the substitution of another.1
It can be asserted with a straight face that the field of continuous optimization has undergone a revolution since 1984 in the sense of the first definition, and that the second
1Merriam–Webster Collegiate Dictionary, Seventh Edition, 1965.
1 definition applies in a philosophical sense: Because the interior-point presence in optimiza- tion today is ubiquitous, it is easy to lose sight of the magnitude and depth of the shifts that have occurred during the past twenty years. Building on the implicit political metaphor of our title, successful revolutions eventually become the status quo. The interior-point revolution, like many other revolutions, includes old ideas that are rediscovered or seen in a different light, along with genuinely new ideas. The stimulating interplay of old and new continues to lead to increased understanding as well as an ever- larger set of techniques for an ever-larger array of problems, familiar (Section 4.4) and heretofore unexplored (Section 5). Because of the vast size of the interior-point literature, it would be impractical to cite even a moderate fraction of the relevant references, but more complete treatments are mentioned throughout. The author regrets the impossibility of citing all important work individually.
2 Linear and Nonlinear Programming: Separated from Birth
Prior to 1984, there was, to first order, no connection between linear and nonlinear pro- gramming. For historical reasons that seem puzzling in retrospect, these topics, one a strict subset of the other, evolved along two essentially disjoint paths. Even more remarkably, this separation was a fully accepted part of the culture of optimization—indeed, it was viewed by some as inherent and unavoidable. For example, in a widely used and highly respected textbook [24] published in 1973, the author comments in the preface that “Part II [unconstrained optimization] . . . is independent of Part I [linear programming]” and that “except in a few isolated sections, this part [constrained optimization] is also independent of Part I”. To provide an accurate reflection of this formerly prevailing viewpoint, we give separate background treatments for linear and nonlinear programming.
2.1 Linear programming 2.1.1 Problem statement and optimality conditions The linear programming (LP) problem involves minimization of a linear (affine) function subject to linear constraints, and can be represented in various mathematically equivalent ways. The two forms of interest here are the all-inequality form,
minimize cTx subject to Ax ≥ b, (1) x and standard form,
minimize cTx subject to Ax = b, x ≥ 0, (2) x where A is m × n. In the standard-form problem (2), the only inequalities are the simple bound constraints x ≥ 0, leading to the crucial (and sometimes overlooked) property that x plays two distinct roles—as the variables and the values of the constraints. It is customary in standard-form problems to assume that A has full rank.
2 ∗ Apointisfeasible if it satisfies the problem constraints. The feasible point x is a ∗ ∗ [H3]. R. Hamilton, “ Formation of singularities in the Ricci flow,” Surveys in Differential solution of the standard-form LP (2) if and only if, for some m-vector y and n-vector z , Geometry 2 (1995), 7-136, International Press. T ∗ ∗ ∗ ∗ ∗ [H4]. R. Hamilton, “The Harnack estimate for the Ricci flow,” Jour. DIff. Geom. 37 c = A y + z ,z≥ 0, and zi xi =0 fori =1,..., n, (3) (1993), 225-243. ∗ ∗ [H5]. R. Hamilton, “Four-manifolds with positive isotropic curvature,” Comm. Anal. where z is the Lagrange multiplier for the bound constraints and y is the Lagrange mul- Geom. 5 (1997), 1-92. tiplier for the equality constraints Ax = b. [M]. J. Milnor, “Groups which act on Sn without Fixed Points,” Amer. Journal of Math. 1957, 623- 630. 2.1.2 The simplex method [P1]. G. Perelman, ‘Spaces with curvature bounded below,” Proceedings of ICM 1994, A fundamental property of linear programs is that, if the optimal objective value is finite, 517-525. a vertex minimizer must exist. (For details about linear programming and its terminology, [P2]. G. Perelman, “The entropy formula for the Ricci flow and its geometric applica- see, e.g., [5], [31], and [19].) The simplex method, invented by George B. Dantzig in 1947, tions,” arXiv.math.DG/0211159. is an iterative procedure for solving LPs that completely depends on this property. The [P3]. G. Perelman, “Ricci flow with surgery on three-manifolds,” arXiv.math.DG/0303109. starting point for the simplex method must be a vertex. Thereafter, every iteration moves [P4]. G. Perelman, “ Finite extinction time for the solutions to the Ricci flow on certain to an adjacent vertex, decreasing the objective as it goes, until an optimal vertex is found. three-manifolds,” arXiv.math.DG/0307245. The underlying motivation for the simplex method is easy to understand, but its simplicity [ST1]. T. Shioya and T. Tamaguchi, “Volume collapsed three-manifolds with a lower is sometimes obscured by a focus on algebraic details. curvature bound,” arXiv.math.DG/0304472. Almost from the beginning, the simplex method (and, by association, linear program- [ST2]. T. Shioya nad T. Tamaguchi, “Collapsing three-manifolds under a lower curva- ming) acquired an array of specialized terminology and notation, such as “basic feasible 56 ture bound,” J. Differential Geom., (2000), 1-66. solution”, “min ratio test”, and the tableau. During the early years of the simplex method, simplex steps were carried out by performing unsafeguarded rank-one updates to the ex- plicit inverse of the square basis matrix. As an aside, use of this risky technique shows that mainstream linear programming was widely separated not only from nonlinear program- ming, but also from numerical linear algebra; fortunately, during the 1960s, the simplex method became more closely connected with state-of-the-art linear algebraic techniques. Although “non-simplex” strategies for LP were suggested and tried from time to time between 1950 and the early 1980s, such techniques never approached the simplex method in overall speed and reliability. Furthermore, the mindset induced by the dominance of the simplex method held sway to such an extent that even techniques labeled as non-simplex were at heart based on the same motivation as the simplex method: to identify the active inequality constraints by staying on a changing subset of exactly-satisfied constraints while reducing the objective function.
2.1.3 Concerns about complexity In practice, because the simplex method routinely and efficiently solved very large linear pro- grams, it retained unquestioned preeminence as the solution method of choice. However, the simplex method was viewed with nagging discontent by those interested in computational complexity, a field whose importance increased during the 1960s and 1970s. An underlying tenet of theoretical computer science is that any “fast” algorithm must be polynomial-time, meaning that the number of arithmetic operations required to solve the problem should be bounded above by a polynomial in the problem size. Although the simplex method almost always converges on real-world problems in a num- ber of iterations that is a small multiple of the problem dimension, it is known that the simplex method can visit every vertex of the feasible region—for example, on the famous 14 3 Klee-Minty “twisted cube” LP; see [31] and [17] for two formulations of this problem. Con- sequently the worst-case complexity of the simplex method is exponential in the problem (dimension X is 0) as limits as t →∞since these pieces would eventually have positive dimension, which means that the simplex method must be a “bad” algorithm. The discon- curvature and hence disappear at finite time. nect between observed speed and theoretical inefficiency was widely known, and there were several claims, subsequently shown to be false, that a provably polynomial-time method for 6 Concluding Remarks LP had been discovered. The first polynomial-time LP algorithm was devised in 1979 by Leonid Khachian of the Perelman’s picture of the effect of Ricci flow with surgery fits perfectly with Thurston’s then Soviet Union, in work that made newspaper headlines around the world. Khachian’s Geometrization Conjecture. The surgeries at finite time implement the connected sum ellipsoid method is based on specialization of general nonlinear approaches developed ear- decomposition into prime factors. This decomposition is necessary before one can hope to lier by other Soviet mathematicians, notably Shor, Yudin and Nemirovskii. In particular, find homogeneous metrics. Then the decomposition along tori and Klein bottles occurs in Khachian’s method does not rely, as the simplex method does, on existence of a vertex analyzing the limits as t →∞. The first decomposition is along incompressible tori into solution or, more generally, the finiteness/combinatorial features of the LP problem. Poly- inflating hyperbolic pieces and collapsing pieces. The second decomposition involves only nomiality of the ellipsoid method arises from two bounds: an outer bound that guarantees the collapsing pieces. It is a decomposition into collapsing pieces whose collapsing limit is existence of an initial (huge) ellipsoid enclosing the solution, and an inner bound that spec- 1-dimension and those whose limit is of 2-dimension. The first are either T 2-bundles over ifies how small the final ellipsoid must be to ensure sufficient closeness to the exact solution. S1 (these are components of the manifolds in the sequence) or T 2 × I. The latter separate See, for example, [31] for details about Khachian’s method. different pieces whose collapsing limit is 2-dimensional, these being Seifert fibered three- Despite its favorable complexity, the performance of the ellipsoid method in practice, manifolds that are collapsing to two-dimensional orbifolds. Thus, we get components that i.e., its actual running time, was extremely slow—much slower than the simplex method. are decomposed along tori into Seifert fibered pieces, components that are Seifert fibered, In fact, in complete contrast to the simplex method, the number of iterations of the ellip- and components that are Solv-manifolds. There is one other possibility which is a collapsing soid method tended to be comparable to its enormous (albeit polynomial) upper bound. component whose limit is a flat three-manifold. Since these are smoothly collapsing to a Thus the simplex method remained “the only game in town” for solving linear programs, smooth limit, the manifolds in the sequence are eventually diffeomorphic to the limit. leading to a puzzling and deeply unsatisfying anomaly in which an exponential-time algo- Notice that the hyperbolic metrics are produced by the Ricci flow, but the other locally rithm was consistently and substantially faster than a polynomial-time algorithm. Even homogeneous metrics are imposed by hand. On the collapsing pieces we obtain topological after Khachian’s breakthrough, the quest continued for an LP algorithm that was not only rather than metric conclusions from the Ricci flow and Alexandrov space theory, but these polynomial, but also efficient in practice.2 The linear programming story will continue in conclusions are strong enough to allow classification of the topological types. Fortunately, Section 3. these are all known to possess locally homogeneous metrics (essentially by induction on dimension). 2.2 Nonlinear programming The three-sphere and other manifolds of finite fundamental group are Seifert fibered. 2.2.1 Problem statement and optimality conditions Thus, they have collapsing metrics with two-dimensional limits. One can ask whether these manifolds occur in the limit as t →∞. In [P3] Perelman argues that in fact this The generic nonlinear programming, or nonlinear optimization, problem involves minimiza- does not happen. Any manifold whose fundamental group is a free product of finite groups tion of a nonlinear function subject to nonlinear constraints. Special cases of nonlinear and infinite cyclic groups must disappear in finite time under the Ricci flow with surgery programming arise when, for example, the objective function is quadratic, the constraints process. are bounds, or the constraints are linear (equalities or inequalities). Here we consider only the all-inequality version of a nonlinear programming problem: 7 References minimize f(x) subject to c(x) ≥ 0, (4) x∈Rn [B].Y. Burago, M. Gromov and G. Perelman, “A. D. Aleksandrov spaces with cruvature where c(x)hasm component functions, and f and {ci} are smooth. (Observe that (4) is bounded below,” Russian Math. Surveys 47 (1992), 1-58. analogous in form to the all-inequality linear program (1).) The n-vector g(x) denotes the [H1]. R. Hamilton, “ Three-manifolds with positive Ricci curvature,” J. Differential gradient of f; the matrix of second partial derivatives will be denoted by H(x). The gradient Geom. 17 (1982), 255-306. and Hessian of ci(x) will be denoted by ai(x)andHi(x). The m×n Jacobian matrix of c(x) [H2]. R. Hamilton, “Non-singular solutions of the Ricci flow on three-manifolds,” T is denoted by A(x), whose ith row is ai(x) .TheLagrangian function associated with (4) is Comm. Analysis and Geometry 7 (1999), 695-729.
2A side comment: Recent work on “smoothed complexity” provides a fascinating explanation of why the simplex method is usually a polynomial-time algorithm; see [32]. 13 4 T L(x, λ)=f(x) − λ c(x), where λ normally represents a vector of Lagrange multipliers. The 5.4 The Collapsed Regions m Hessian of the Lagrangian with respect to x, denoted by W ,isW (x, λ)=H− j=1 λjHj(x). ≥ At this point we use no more information from the Ricci flow. The argument now turns to The constraint ci(x) 0issaidtobeactive atx ¯ if ci(¯x) = 0 and inactive if ci(¯x) > 0. Let Aˆ(x) denote the Jacobian of the active constraints at x,andletN(x)denoteamatrix results from the theory of collapsed manifolds. In our situation, we have a sequence Mt n ˆ of 3-dimensional manifolds that are collapsing with (local) lower curvature bounds. The whose columns form a basis for the null space of A. Throughout the remainder of the paper, we assume the following conditions, which are idea is to pass to a limit X (of a subsequence) which is a more general metric space than ∗ a riemannian manifold, but one in which the notion of curvature being bounded below still sufficient to ensure that x is an isolated constrained minimizer of (4): ∗ ∗ makes sense. The natural category for such objects is the category of Alexandrov spaces, 1. c(x ) ≥ 0andAˆ(x )hasfullrank; see [B] and [P1]. Such spaces have a uniform dimension. In the case of limits the dimension ∗ ∗ T ∗ ∗ of the limit is most that of the manifolds in the sequence. The analysis the depends on the 2. g(x )=A(x ) λ ,withλj ≥ 0and dimension of the limiting Alexandrov space X. Let us first analyze the case of a sequence of ∗ ∗ 3-manifolds converges with (global) curvature bounds to an Alexandrov space X, cf [ST1, λj cj(x )=0,j=1,...,m;(5) ST2]. ∗ ∗ 3. λj > 0ifcj(x )=0,j =1,..., m; 5.4.1 X has dimension 0 ∗ ∗ ∗ ∗ 4. N(x )T W (x ,λ )N(x ), the reduced Hessian of the Lagrangian, is positive definite. This is the case when the manifold (or one of its components) is collapsing to a point. In Relation (5), that each pairwise product of constraint and multiplier must be zero, is called this case, the manifolds in the sequence eventually have positive curvature. These satisfy ∗ ∗ complementarity. Condition 3, strict complementarity, requires that one of cj(x )andλ the Geometrization Conjecture. j must be positive. 5.4.2 X has dimension 1 2.2.2 Newton’s method In this case X is either a circle or an interval and the manifolds M locally fibers over X Newton’s method occurs in multiple forms throughout optimization. When solving the with two-torus fibers. Hence, these manifolds are Solv-manifolds or are T 2 × I. Of course, nonlinear equations Φ(z)=0,letJ(z) denote the Jacobian matrix of Φ. If zk is the current the Solv-manifolds satisfy the Geometrization Conjecture. point and J(zk) is nonsingular, the Newton step δk is the solution of the linear system
5.4.3 X has dimension 2 J(zk)δk = −Φ(zk), (6) In this case the manifolds M ‘fibers’ over X with circle fibers. The quotation marks reflect so that δk is the step from zk to a zero of the local affine Taylor-series model of Φ. the fact that the structure is that of a Seifert fibration over a two-dimensional orbifold. For unconstrained minimization of f(x) starting from xk,theNewtonsteppk is designed T to minimize a local Taylor-series quadratic model of f(xk + p), namely f(xk)+g(xk) p + 5.4.4 has dimension 1 T X 3 2 p H(xk)p. If the current Hessian H(xk) is positive definite, pk solves the linear system This case can be quite delicate since three-dimensional Alexandrov spaces are quite compli- H(xk)p = −g(xk). (7) cated. But in this case the Ricci flow gives us more. The parabolic nature of the equation tells us that curvature bounds imply bounds on all derivatives of curvature. Hence, in When minimizing f(x) subject to m linear equality constraints Ax = b,theNewtonstep this case the limit is actually a smooth riemannian manifold not a more general Alexan- pk should minimize the local Taylor-series quadratic model of f subject to also satisfying drov space. This manifold is automatically flat, and hence satisfies the Geometrization the constraints A(xk + pk)=b,sothatpk is a solution of the quadratic program Conjecture. 1 T T minimize p Hkp + gkp subject to Ap = b − Axk, (8) Now we must adapt this limiting process where we have a global lower bound to cur- p∈Rn 2 vature to our situation where the curvature lower bound is only a local one. Perelman’s claim is that one obtains a further decomposition of the manifolds in the sequence along where Hk = H(xk)andgk = g(xk). Under appropriate conditions, pk and a “new” multi- incompressible tori into pieces of the above types, i.e., into pieces satisfying the conclusion plier yk+1 satisfy the following n + m linear equations: of the Geometrization Conjecture. Notice that one will never see pieces of the first type T Hk A pk −gk = , (9) A 0 −yk+1 b − Axk 12 5 where yk+1 is an estimate of the Lagrange multipliers for the equality constraints. The T The second type of operation is meaningful metrically since the balls in question have dif- matrix in (9) is nonsingular if A has full rank and the reduced Hessian NA HkNA is positive definite, where NA is a basis for the null space of A.IfAxk = b, the second equation in (9) ferent metrics, but topologically it does nothing. The last type of operation may change the becomes Apk = 0, implying that pk must lie in the null space of A. topology significantly since entire components are removed, but, as we have seen, each of A “pure” Newton method for zero-finding begins with an initial point z0, and generates these components satisfies the conclusion of the Geometrization Conjecture, so that we are a sequence of Newton iterates {zk},wherezk+1 = zk + δk,withδk defined by (6), and removing pieces of the prime decomposition of the original three-manifold that do satisfy similarly for minimization, using (7) and (9). Under various conditions that can be quite the conjecture and keeping those about which we have not yet learned anything. restrictive, a pure Newton method converges quadratically to a solution. One way to encourage convergence from a general starting point is to perform a line 5.3 Limits at infinity search in which the new iterate is defined by zk+1 = zk +αkδk, where the positive scalar αk is The next step in Perelman’s argument is to analyze the limits of the Ricci flow with surgery chosen to decrease a merit function that measures progress. In unconstrained optimization, as t →∞. Once again in this part of the argument he is following the path initiated by the merit function is typically the objective function. Standard line search acceptance Hamilton in [H2], though his situation is more complicated for two reasons – (i) he has criteria that ensure convergence are discussed in, for example, [29, 28]. A second strategy Ricci flow with surgeries instead of a Ricci flow and (ii) he does not make the curvature is based on defining a trust region around the current iterate within which the local model bound assumption as Hamilton did. In spite of these complications Perelman claims that can be trusted. In optimization, the step in a trust-region method is typically chosen much of Hamilton’s analysis can be adapted. He claims: to minimize (approximately) the local Taylor-series quadratic model subject to remaining within a (normally 2) trust region. Claim 5.3.1. • There is a finite collection of complete hyperbolic manifolds Hi of finite volume and for all sufficiently large t an embedding 2.2.3 Barrier methods for constrained optimization ϕt : Hi → Mt The 1960s were the heyday of unconstrained optimization, and, as a result, it was common i practice to convert constrained problems into unconstrained subproblems or sequences of which for sufficiently large t is arbitrarily close to an isometry on an arbitrarily large unconstrained subproblems. Penalty and barrier methods were especially popular, both compact subset of Hi provided that we rescale the metric on Mt by c/t for an motivated by minimizing a composite function that reflects the original objective function i appropriate constant c independent of t. as well as the influence of the constraints. Modern interior methods are closely related to “classical” (1960s) barrier methods, which we now describe. • For all t>>1 the boundary tori of the Hi are incompressible tori in Mt. The logarithmic barrier function associated with the problem (4) is • For all t>>1, the complement of the image of a sufficiently large compact subset of m i Hi has a metric which is arbitrarily collapsed with lower curvature bounds. B(x, µ)=f(x) − µ ln cj(x), (10) j=1 For a point x in a complete riemannian manifold define r(x) to be the supremum of 2 r ≥ 0 such that the Rm(y) ≥−r on the metric ball Br(x)ofradiusr centered at x. where µ is a positive scalar called the barrier parameter. The logarithmic terms are well To say that a metric is w-collapsed at a point x with lower curvature bounds means that defined at points x for which c(x) > 0, but become unbounded above as x approaches any 3 Vol(Br(x)(x))
2 1 3 3 3 3 Since gB (xµ)vanishes,xµ can be interpreted as a highly special point at which the objec- • a closed manifold diffeomorphic to S × S , RP , S or RP #RP . tive gradient is a nonnegative linear combination of the constraint gradients. Further, the 2 coefficients in the linear combination have a specific relationship with µ and the constraint • a tube diffeomorphic to S × I covered by regions where the metric is very close to values, i.e., a standard round metric of curvature√ Q ≥ Q on the sphere times a interval whose m µ T length is a large multiple of 1/ Q . g(xµ)= aj(xµ)=A (xµ)λµ(xµ), (16) cj(xµ) j=1 • a region double covered by a tube as in the previous case. where the multiplier estimate λµ satisfies • a manifold diffeomorphic to D3 with a neighborhood of the boundary of the third type. −1 λµ(x)=µC(x) 1. (17) Furthermore, in the first case the metric when rescaled to have a fixed diameter is converging to a metric of constant positive curvature. A rearranged component-wise version of (17) is Thus, in the first two cases the manifold becomes completely singular at the limiting λµ(xµ) ici(xµ)=µ. (18) time and admits a metric homogeneous metric of non-negative sectional curvatures. That ∗ ∗ ∗ is to say the manifold satisfies the conclusion of the Geometrization Conjecture. Thus, we This relationship is similar to complementarity (5), which holds at x between λ and c(x ), can assume that manifolds of these types do occur we can remove them at the singular time, and is sometimes called perturbed complementarity or the centering property of xµ. knowing that these satisfy the conclusion of the Geometrization Conjecture. We are left To move from the current point x to xµ, a straightforward strategy is to apply Newton’s to consider tubes, regions double covered by tubes and regions diffeomorphic to D3 with a method to a local quadratic model of the barrier function. Omitting arguments, the resulting neighborhood of the boundary being a tube. We assume (for simplicity of the exposition n × n primal Newton barrier equations are only) that the manifold in question admits no RP 2, then the only cases that we are left to T −1 consider are tubes and three-balls with tubes as neighborhoods of the boundary. We call HB p = −g + µA C 1, (19) the latter caps. where “primal” refers to the original problem variables x. Although barrier methods were widely used during the 1960s, they suffered a severe 5.2 The Surgery Process decline in popularity in the 1970s for various reasons, including perceived inefficiency com- pared to alternative strategies and worries about inherent ill-conditioning. With respect To treat singularities occurring inside of tubes and caps, Perelman introduces the notion of to the latter, it was observed in the late 1960’s (see [23, 25]) that, if 1 ≤ m
9 8 3 The Revolution Begins Theorem 4.2.1. (Hamilton) Let X3 be a compact connected three-manifold with non- negative Ricci curvature. Then one of the following happens: 3.1 Karmarkar’s method • The Ricci curvature becomes strictly positive for all t>0 sufficiently small. In In 1984, Narendra Karmarkar [21] announced a polynomial-time LP method for which he this case, the Ricci flow develops a singularity in finite time. As the singularity reported solution times that were consistently 50 times faster than the simplex method. develops the diameter of the manifold is going to zero. Rescaling the evolving family This event, which received publicity around the world throughout the popular press and of metrics so that their diameter is 1 leads to a family of metrics converging to a media, marks the beginning of the interior-point revolution. metric of constant positive curvature. In particular, the manifold is diffeomorphic to Karmarkar’s method had several unusual properties: a special, non-standard form was 3 the quotient of S by a free orthogonal action of a finite group. assumed for the linear program; nonlinear projective geometry was used in its description; and no information was available about the implementation. Amid the frenzy of interest in • The manifold is finitely covered by a metric product of a compact surface of positive Karmarkar’s method, it was shown in 1985 (and published the next year [15]) that there curvature and S1. Again the Ricci flow develops a singularity in finite time, and was a formal equivalence between Karmarkar’s method and the classical logarithmic barrier the manifold in question is diffeomorphic to S2 × S1, or is finitely covered by such a method applied to the LP problem. Soon researchers began to view once-discarded barrier manifold. methods in a previously unthinkable context: as the source of polynomial-time algorithms • The metric is flat and the evolution equation is constant. In this case, of course, the for linear programming. manifold is finitely covered by T 3. For several years the tie between Karmarkar’s method and barrier methods was con- tentious and controversial. Researchers argued about whether the two approaches were In [H2] Hamilton went on to analyze under certain extra assumptions what happens to fundamentally different, or very similar, or something in between. Now that the dust has the metric as t →∞. settled, derivations of interior methods typically involve barrier functions or their properties, Theorem 4.2.2. (Hamilton) Suppose that the Ricci flow on a compact riemannian 3- such as perturbed complementarity (18). Readers interested in Karmarkar’s method should manifold M exists for all t ∈ [0, ∞) and that the normalized curvature Rm(x, t)·t is bounded consult his original paper [21], or any of the many comprehensive treatments published as t →∞. Then there is a finite set of complete hyperbolic manifolds Hi of finite volume since 1984 (e.g., [19, 30, 41, 35, 44]). → and for each t>>1 an embedding ϕt : i Hi M which are converging on each compact Beyond establishing the formal connection between Karmarkar’s method and barrier subset of i Hi to an isometry provided that we rescale the metric at time t byaconstant methods, [15] reported computational results comparing a state-of-the-art (in 1985) simplex dependent on t. The tori in the Hi near infinity are incompressible tori in M and the code, MINOS [26], and an implementation of the primal Newton barrier method on a widely complement of the image of ϕt has a decomposition along incompressible tori and Klein available set of test problems. To the astonishment of many who believed that nothing could bottle into pieces each of which admits a locally homogeneous metric of finite volume. beat the simplex method, the barrier method was faster on several of the problems, and competitive on many others. Thus, under two addition hypothesis – existence of the Ricci flow for all time and At this stage, the interior-point revolution gathered momentum and accelerated in sev- a bound on the normalized curvature – Hamilton established the conclusion of the Ge- eral directions, to be described in Sections 4 and 5. First, however, we describe the derivation ometrization Conjecture. of the primal Newton barrier method for LP given in [15].
5 Perelman’s Claims 3.2 The primal Newton barrier method for LP
In a sequence of three manuscripts [P2, P3, P4] posted on the math archive in the last 14 To make the connection between linear programming and a barrier method, consider a T months Perelman has given arguments for a sequence of claims about the Ricci flow and its standard-form linear program—minimize c x subject to Ax = b and x ≥ 0—with three relation to the Geometrization Conjecture. The overriding claim is that Hamilton’s program properties: (a) the set of x satisfying Ax = b and x>0 is nonempty; (b) the set (y,z) T can be completed without the extra hypotheses he imposed and hence Geometrization satisfying A y + z = c and z>0 is nonempty; and (c) rank(A)=m. Because the only Conjecture is true and can be proved using the Ricci flow, generalized to allow surgeries. inequality constraints are the bounds x ≥ 0, the associated logarithmic barrier function (see (10)) is n T 5.1 Singularities at finite time B(x, µ)=c x − µ ln xj, (20) Perelman’s first result is about the nature of the singularities that develop in finite time j=1 under Ricci flow.
8 9 and the barrier subproblem is to minimize (20) subject to satisfying the equalities Ax = b: introduces the curvature operator which is a section of the second exterior power of the n cotangent bundle with values in the orthogonal endomorphisms of the tangent bundle. One T − minimize c x µ ln xj subject to Ax = b. (21) then defines the curvature tensor j=1 Rm(X, Y, Z, W)=−R(X, Y )Z, W. The gradient and Hessian of the barrier function (20) have particularly simple forms: This is a section of the fourth exterior power of the cotangent bundle. It is skew in the − −1 −2 gB = c µX e and HB = µX . (22) first two variables and skew in the last two variable and symmetric under interchange of variables (1, 2) with variables (3, 4). Thus, we can view it as a symmetric tensor on the The barrier subproblem (21) has a unique minimizer if (b) is satisfied. At the optimal second exterior power of the cotangent bundle. The Ricci curvature is the trace on Rm on solution of (21), there exists y such that the middle two variables. Thus, if e1,...,en is an orthonormal frame at a point we have −1 T T −1 gB (x, µ)=c − µX 1 = A y, so that c = A y + µX 1. (23) n Ric(ei,ej)= Rm(ei,ek,ek,ej). The central path (barrier trajectory) for a standard-form LP is defined by vectors xµ and k=1 yµ satisfying This is a contravariant symmetric two-tensor on the manifold. In local coordinates, its Axµ = b, xµ > 0; leading term is a linear expression in the second partial derivatives of the entries of the T −1 matrix gij describing the metric. A yµ + µXµ 1 = c. (24)
The central path has numerous properties of interest; see, e.g., [20], [19],[35],[41], and [44]. 4.2 The Ricci Flow Equation Assume that we are given a point x>0forwhichAx = b. Using (22), the Newton Hamilton introduced the Ricci Flow equation: equations (9) for problem (21) are g(t)=−2Ric(g(t)). µX−2 AT p −c + µX−11 = , (25) A 0 −y 0 To indicate the analogy with non-linear versions of the heat equation let us write the evolution equation for the scalar curvature R (which by definition is the trace of the Ricci so that the Newton step p in x satisfies curvature). The equation is
−2 −1 T 2 µX p + c − µX 1 = A y (26) R(t)=∆R(t)+ R2(t)+|Ric0(t)|2, 3 2 for some Lagrange multiplier vector y. Multiplying (26) by AX and using the relation where Ric0 is the traceless part of the Ricci curvature. It is clear from this expression that Ap = 0 to eliminate p,weobtain the minus sign in the original equation is forced on us – without it we would have non-linear AX2ATy = AX2c − µAXe = AX(Xc − µ1). (27) versions of the backwards heat equation which is an ill posed equation. The factor of 2 is for convenience only. Since A has full rank and x>0, the matrix AX2AT is positive definite, so that (27) has a In [H1, H3, H4] Hamilton proved: unique solution y. Using (26), p is defined in terms of y as • Short-time existence. If g0 is a smooth metric then there is some >0 depending on 1 g0 and a solution to the Ricci flow equation defined for t ∈ [0,)withg(0) = g0. p = x + X2(ATy − c). µ • If the solution exists on the time interval [0,T) but does not extend to any strictly Because Ax = b and Ap = 0, the new point x + αp will continue to satisfy the equality larger time interval, then there is a point x in the manifold for which curvature tensor constraints for any α. However, α may need to be less than one in order to retain strict Rm(x, t) of the metric g(t) is unbounded as t approaches T . feasibility with respect to the bound constraints. In [H1] Hamilton also analyzed manifolds the Ricci flow on 3-manifolds with non- negative Ricci.
7 10 4 The Revolution Advances Notice by Corollary 2.2.1 each torus boundary component has fundamental group in- jecting into each of the three-manifolds that it bounds. It follows from Van Kampen’s Following the announcements of Karmarkar’s method and its connection with the logarith- theorem, that the fundamental group of each torus injects into π1(M). Such tori are called mic barrier method, researchers began to develop other interior LP methods with improved incompressible tori. More generally, a surface of genus at least one in a three-manifold is complexity bounds, and to derive properties of barrier methods applied to linear programs. said to be incompressible if its fundamental group injects into the fundamental group of the Furthermore, since barrier methods (unlike the simplex method) were originally intended three-manifold. It follows immediately that the Klein bottles also are incompressible. Thus, for nonlinear problems, it was evident that they could be applied not just to linear pro- one can formulate the Geometrization Conjecture as saying that there is a decomposition gramming, but to other optimization problems, such as quadratic programming, linear and of M along incompressible tori and Klein bottles into pieces whose interiors admit finite nonlinear complementarity, and nonlinear programming. volume complete geometric structures. 4.1 A change in perspective 3.3 Relation to the Poincar´e Conjecture The interior-point revolution has led to a fundamental shift in thinking about continuous Suppose that we have a prime homotopy three-sphere Σ that satisfies the conclusion of the optimization. Today, in complete contrast to the era before 1984, researchers view linear geometrization conjecture. Since π1(Σ) = {1}, Σ has no incompressible tori and hence the and nonlinear programming from a unified perspective; the magnitude of this change can decomposition of Σ referred to in Conjecture 3.2.1 must be trivial. That is to say Σ has be seen simply by noting that no one would seriously argue today that linear programming a locally homogeneous metric. Again since π1(Σ) is trivial, the homogeneous model for is independent of nonlinear programming. Σ must itself be compact and Σ is diffeomorphic to the model space. The only compact Beyond a broadened perspective, one might wonder whether the revolution has made a three-dimensional model is S3, and consequently, Σ is diffeomorphic to S3. Notice that substantive difference: is the net result simply that the log barrier method was rediscovered this argument (except for the very last step) applies equally well to prime 3-manifolds of and applied to new problems? The answer to this is an emphatic “No”. As we shall finite fundamental group. The conclusion is that the Geometrization Conjecture for such try to indicate in the remainder of the paper, there have been fundamental advances in manifolds implies that they are space-forms, i.e., are quotients of S3 by a subgroup of SO(4) complexity theory, algorithms, linear algebra, and solvable problems, all as a result of the acting freely. Such groups were classified by Hopf, see [M]. interior revolution.
4.2 Complexity 4 The Ricci Flow A signature of interior methods is the existence of continuously parameterized families of Let us now introduce Hamilton’s Ricci flow, cf [H1]. This is a parabolic evolution equation approximate solutions that asymptotically converge to the exact solution; see, for example, a riemannian metric on a manifold. As we shall see it is a non-linear analogue of the heat [20]. As the parameter approaches its limit, these paths trace smooth trajectories with equation for the metric tensor. Its general form is geometric properties (such as being “centered” in a precisely defined sense) that can be analyzed and exploited algorithmically. These paths also play a critical role in complexity g (t)=F (g(t)), analyses of interior algorithms. where a solution is a one-parameter family of metrics whose time derivative is given by a The elements in a typical proof of polynomial complexity for an interior method are: functional F (g(t)). This functional must be of the same tensor type as the metric, i.e., • Characterizing acceptable closeness to the solution through a stopping rule. Such a contravariant symmetric two-tensor and should (we hope) involve no more than second a rule is needed because an interior method that generates strictly feasible iterates derivatives of the metric. It must also be natural (i.e., independent of the coordinates used cannot produce, within a finite number of iterations, a solution that lies exactly on a to write the equation locally). There are not too many choices for such an F , and the one constraint; Hamilton introduced is F equal to a constant multiple of the Ricci curvature on the metric. • Defining a computable measure of closeness to the parameterized path associated with 4.1 Curvature the problem and the algorithm; Recall that given a riemannian metric, there is a unique symmetric connection ∇ on the • Showing that a Newton step, or a suitably small number of Newton steps, taken from tangent bundle of the manifold. Defining a point close to the path will stay sufficiently close to the path; • R(X, Y )=∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] Decreasing the controlling parameter at a rate that allows a polynomial upper bound on the number of iterations needed to become close enough to the solution. 6 11 Innumerable papers have been written about complexity issues in interior methods; the surveys [19, 30, 35, 44] (among others) provide details and further references. N.B. To reverse the process and reconstruct the manifold from its prime factors one Every discussion of the analysis of interior methods should pay tribute to the work of needs to work with oriented manifolds and oriented connected sums. That is to say two Nesterov and Nemirovskii, whose work in the late 1980s extended the scope of polynomial- non-diffeomorphic three-manifolds can have the same prime factors. 2 time complexity results to a wide family of convex optimization problems; for details, see An essential S ⊂ X is a two-sphere that does not bound a three-ball in X. If this sphere [27]. One of their major contributions was to define self-concordant barrier functions.A separates, then cutting X along it and filling in the holes with three-balls implements a 2 convex function φ from a convex region F 0 ∈Rn to R is κ-self-concordant in F 0 if (i) φ connected sum decomposition of X. If on the other hand a S ⊂ X does not separate, then 2 1 is three times continuously differentiable in F 0 and (ii) for all y ∈F0 and all h ∈Rn,the this procedure has the effect of removing from X a prime factor diffeomorphic to S × S . following inequality holds: Because of the prime decomposition, all questions about the topology of three-manifolds can be reduced to questions about prime three-manifolds. For example: 3/2 |∇3φ(y)[h, h, h] |≤2κ hT ∇2φ(y)h , Corollary 3.1.3. Σ3 is a homotopy three-sphere if and only if all its prime factors are where ∇3φ(y)[h, h, h] denotes the third differential of φ at y and h. The logarithmic barrier homotopy three-spheres. Thus, there is a counterexample to the Poincar´e conjecture if and functions associated with linear and convex quadratic programming are self-concordant only if there is a prime counterexample. with κ = 1. Existence of a self-concordant barrier function for a convex problem is closely 2 By the sphere theorem, if π2(M) = 0, then X contains an essential S . related to existence of polynomial-time algorithms. Using the concept of self-concordance, new barrier functions have been devised for certain convex programming problems, such as semidefinite programming, that were previously considered computationally intractable; 3.2 Thurston’s Geometrization Conjecture see Section 5. Excluding manifolds with locally homogeneous metrics based on S2 × R, all manifolds with Despite the polynomial bounds typically associated with interior methods, a mystery locally homogeneous metrics have universal coverings which are diffeomorphic either to R3 remains similar to that still holding for the simplex method: interior methods almost invari- or S3. This means that every S2 in the universal cover bounds a three-ball. The same is ably require a number of iterations that is much smaller than the (very large) polynomial then true on the original manifold. Thus, the only 3-manifolds with locally homogeneous upper bound. The reasons for these disparities are not yet understood, but perhaps one metrics and essential S2’s are S2 × S1 and RP 3#RP 3, the latter being the only example day they will be. of a non-prime manifold admitting a locally homogeneous metric. Thurston’s geometrization conjecture is about prime manifolds. For simplicity we re- 4.3 Barrier methods revisited strict attention to orientable manifolds. The problem of ill-conditioning, as noted earlier, has haunted interior methods since the Conjecture 3.2.1. (Thurston’s Geometrization Conjecture) Let M be a compact, ori- late 1960s, but there has been substantial recent progress in understanding this issue. A entable, prime three-manifold. Then there is an embedding of a disjoint union of 2-tori detailed analysis was given in [37] of the structure of the primal barrier Hessian (15) in an 2 and Klein bottles T ⊂ M such that every component of the complement admits a lo- entire neighborhood of the solution. Several papers ([13], [11], [40], [42]) have analyzed the i i cally homogeneous riemannian metric of finite volume. stability of specific factorizations for various interior methods. Very recently, the (at first) surprising result was obtained ([39], [43]) that, under con- The locally homogeneous metrics on the various components of this cutting process can be ditions normally holding in practice, ill-conditioning of certain key matrices in interior modelled on different homogeneous manifolds. methods for nonlinear programming does not noticeably degrade the accuracy of the com- Since we are working with orientable manifolds, the boundary of a neighborhood of a puted search directions. In particular, in modern primal-dual methods (see Section 4.4.1), if Klein bottle is a two-torus, so that the resulting geometric pieces are interiors of compact a backward-stable method is used to solve the condensed primal-dual system (the analogue manifolds all of whose boundary components are tori. of the barrier Hessian), the computed solution has essentially the same accuracy as that of N.B. There can in general be more than one such family of tori and Klein bottles up to the well-conditioned full primal-dual system (33). However, this result crucially depends on isotopy for which the conclusion of the geometrization conjecture holds. For example, let 1 2 1 the special structure of the relevant ill-conditioned matrices, in particular their asymptotic Σ2 be a surface of genus 2. Then Σ2 × S has a geometric structure. Let T ⊂ Σ2 × S be ˆ relationship with A and the reduced Hessian of the Lagrangian. A similar kind of analysis the torus lying over a loop in Σ2 separating it into two once-punctured tori. Then each of 1 applies to the search direction computed with the primal barrier Hessian. Consequently, the two components of Σ2 × S \ T also has a geometric structure. ill-conditioning in interior methods undeniably exists, but will tend to be benign. To obtain uniqueness one takes a family in X satisfying the conclusion with a minimal It turns out that ill-conditioning is not the only defect of primal barrier methods. Even number of components. With this condition, the family is unique up to isotopy. if the Newton direction is calculated with perfect accuracy, primal barrier methods suffer from inherently poor scaling of the search direction during the early iterations following a 5 12 reduction of the barrier parameter; see [38, 6]. Thus, unless special precautions are taken, a Locally homogeneous examples modelled on this group are called solv-manifolds. full Newton step cannot be taken immediately after the barrier parameter is reduced. This fundamentally undesirable property implies that the classical primal barrier method will be • R R G = PSL2( ), the universal covering group of PSL2( ). This manifold can also be unavoidably inefficient. H2 viewed as the universal metric covering of the unit tangent bundle to . A fascinating but unresolvable question is whether the loss of popularity of barrier Finite volume nil-manifolds are compact and are circle bundles over T 2 with non-zero methods in the 1970s was unfairly blamed on ill-conditioning (which is often a genuine Euler number, or are finitely covered by such circle bundles. Finite volume Solv-manifolds villain); the observed inefficiencies were probably attributable to the just-mentioned flaw of fiber over S1 with T 2-fiber or have a two-sheeted covering of this type. The monodromy of the primal barrier method rather than to ill-conditioning. this fibration is given by an element of SL2(Z) whose eigenvalues are real and have absolute 4.4 New algorithms for old problems value different from 1. Geometric structures based on PSL2(R) are Seifert fibered manifolds over hyperbolic two-dimensional orbifolds of finite area. The non-compact examples are all Leading candidates for the most popular algorithms to emerge from the interior revolution diffeomorphic to interiors of compact three-manifolds all of whose boundary components belong to the primal-dual family. Although there is no precise, universally accepted defini- are tori. tion of a primal-dual method, these methods are almost always based on applying Newton’s Examination of all the possibilities shows us the following: method to nonlinear equations stated in terms of the original (“primal”) problem variables, along with “dual” variables representing the Lagrange multipliers. Corollary 2.2.1. If X3 is orientable and admits a locally homogeneous riemannian metric of finite volume, then X is diffeomorphic to the interior of a compact three-manifold with 4.4.1 Primal-dual methods for linear programming boundary, all of whose boundary components are tori. Furthermore, each of these tori has fundamental group which injects into the fundamental group of X. The only geometric The optimal solution x of the barrier subproblem (21) for a standard-form LP satisfies the T −1 structures with finite volume but non-compact examples are those modelled on H3, H2 × R condition c = A y + µX 1 for some m-vector y (see (23)). Defining the n-vector z as −1 and PSL2(R). The manifolds of finite volume of these types are either hyperbolic three- µX 1, we may replace this condition by the following two equations: manifolds or are Seifert-fibered with hyperbolic two-dimensional base. c = ATy + z and Xz = µe. (28)
3 Topology of Three-manifolds The second relation in (28) has a clear resemblance to the perturbed complementarity condition (18) that holds along the barrier trajectory between the inequality constraints There are several reasons that, unlike the case in dimension two, not every compact three- (here, the variables x) and Lagrange multiplier estimates. manifold can admit a homogeneous metric of finite volume. The most obvious reason The primal Newton barrier algorithm described in Section 3.2 is formulated in terms of has to do with the fact that three-manifolds are not necessarily prime, whereas, with one only primal variables x; the Lagrange multiplier estimate y of (25) arises as a byproduct of exception, all locally homogeneous manifolds are prime. the equality-constrained Newton subproblem. One could alternatively seek primal variables x and dual variables y (for the equalities) and z (for the inequalities) satisfying the central- 3.1 The prime decomposition path conditions (24) rewritten to include z: 3 Definition 3.1.1. A three-manifold X is said to be prime if it is not diffeomorphic to S Ax = b, x > 0,ATy + z = c, z > 0, and Xz = µ1. (29) and if every S2 ⊂ X that separates X into two pieces has the property that one of these two pieces is a three-ball. Equivalently, anytime we write X asaconnectedsumoftwo Note that only the third equation in (29) is nonlinear. manifolds one of them must be the three-sphere. Applying Newton’s method (6) to these 2n+m equations, we obtain the following linear system for Newton steps in x, y,andz: One of the first theorems in the topology of three-manifolds is due to Knesser: A 00 px b − Ax Theorem 3.1.2. Every three-manifold admits a decomposition as a connected sum of prime T T 0 A I py = c − A y − z . (30) three-manifolds, called prime factors. This decomposition is unique up to the order of the prime factors (and diffeomorphism of the factors). Z 0 X pz µe − XZe
Eliminating pz and px gives the linear system
−1 T −1 −1 T AZ XA py = AZ X(c − µX e − A y)+b − Ax, (31) 4 13 where AZ−1XAT is symmetric and positive definite, with the form AD2AT for a nonsingular • R3 diagonal matrix D.Oncepy is known, pz and px may be calculated directly from the second , which is flat. and third block rows of (30) without solving any equations. 3 • H of constant curvature −1. Primal-dual methods for linear programming have been enormously successful in prac- tice. For a detailed discussion of many aspects of primal-dual methods, see [41]. Any manifold with a geometric structure modelled on the first is a riemannian manifold A striking effect of the interior revolution has been the magnitude and extent of perfor- of constant positive sectional curvature. These are of the form S3/Γ where Γ is a finite mance improvements in the simplex method, which was (wrongly) thought in 1984 to have subgroup of SO(4) acting freely on S3 and include S3, RP 3, lens spaces, as well as the already reached its speed limit. In LP today, interior methods are faster than simplex for quotients by the symmetry groups of the exceptional regular solids. some very large problems, the reverse is true for some problems, and the two approaches are Any geometric structure of finite volume modelled on R3 has a flat metric, and hence more or less comparable on others; see [2]. Consequently, commercial LP codes routinely is finitely covered by a flat T 3. In particular, these manifolds are all compact. offer both options. Further analysis is still needed of the problem characteristics that deter- A geometric structure modelled on H3 is a complete, finite volume hyperbolic manifold, mine which approach is more appropriate. Unless a drastic change occurs, both approaches i.e., a riemannian manifold with all sectional curvatures equal to −1 that is complete and are likely to remain viable into the foreseeable future. of finite volume. There are infinitely many such manifolds up to diffeomorphism, both compact and non-compact. By Mostow rigidity, a three-manifold admits at most one (up 4.4.2 Primal-dual methods for nonlinear programming to isomorphism) complete, finite volume hyperbolic metric. If the manifold in question is not compact then each component of a neighborhood of infinity is diffeomorphic to Because of inherent flaws in primal barrier methods (see Section 4.3), primal-dual methods T 2 ×(1, ∞). Geometrically the torus sections become small exponentially fast as t increases. based on properties of xµ are increasingly popular for solving general nonlinear programming In particular, any finite volume hyperbolic three-manifold is diffeomorphic to the interior of problems; see, for example, the recent papers [8, 4, 11, 7, 14]. As in primal-dual methods a compact three-manifold with the property that every boundary component is a two-torus. for LP, the original (primal) variables x and the dual variables λ (representing the Lagrange Next we have the reducible examples: multipliers) are treated as independent. The usual motivation for primal-dual methods is to find (x, λ) satisfying the equations • S2 × R. that hold at xµ. In the spirit of (16) and (17), (xµ,λµ(xµ)) satisfy the following n + m 2 nonlinear equations: • H × R. T g = A λ and ciλi = µ, i =1,...,m. (32) Finite volume geometric structures based on S2 × R are automatically compact and are 2 1 Applying Newton’s method, we obtain the (full) n + m primal-dual equations for Newton either isometric to S × S or have this riemannian manifold as the double covering. In the 3 3 steps in x and λ: later case, the manifold is diffeomorphic to RP #RP . Finite volume geometric structures T T H2 × R × 1 W −A px −g + A λ based on are of the form Σ S where Σ is a finite area hyperbolic surface or are = (33) finitely covered by such manifolds. In the later case, the manifold is Seifert fibered over ΛA C pλ µ1 − Cλ a hyperbolic two-dimensional orbifold of finite area. The examples Seifert fibered over a where W is the Hessian of the Lagrangian evaluated at (x, λ). hyperbolic base can be non-compact, but if they are then they are diffeomorphic to interiors All primal-dual methods are based on more or less the idea just described, which is of compact manifolds with every boundary component being a torus. sometimes presented in terms of the logarithmic barrier function (hence leading to properties Lastly, we have the examples where M is a connected Lie group with a left-invariant of xµ), or else in terms of perturbed complementarity (18) as a desired property in itself. metric. Then the group itself is embedded in the group of isometries of M which can be of Naturally, the equations (33) do not begin to constitute a complete algorithm for nonlinear larger dimension. Three-dimensional examples of this type admitting locally homogeneous programming. Primal-dual methods are the object of active research today, and span a examples of finite volume are: wide range of approaches to algorithmic details, including • The unipotent group of strictly upper triangular three-by-three matrices N 3. Locally 1. formulation of the constraints; homogeneous manifolds modelled on this group are called nil-manifolds.
2. solution of the linear system that defines the Newton steps; • The solvable group which is written as a semi-direct product R2 R∗ where the action of t ∈ R∗ on R2 is diagonal and given by the matrix 3. treatment of indefiniteness; et 0 4. strategies for encouraging progress toward the solution from an arbitrary starting . 0 e−t point;
3 14 5. treatment of equality constraints (an option needed for a general-purpose nonlinear complete the program and establish the Geometrization Conjecture and hence the Poincar´e programming method). conjecture. Perelman’s arguments are quite intricate and involve many strikingly original ideas. The mathematical community is still trying to digest the argument and ascertain 4.5 Linear algebra whether it is indeed a complete and correct argument. So far it is holding up under this scrutiny. At this point one can say with assurance that Perelman has made tremendous Interior methods would not be fast or reliable without efficient, numerically stable linear al- advances in understanding the nature of Ricci flow. gebraic techniques for solving the associated distinctive, specially structured linear systems. Great advances have taken place since 1984 in sparse Cholesky-based techniques for fac- torizing matrices of the form AT D2A,whereD is diagonal and is becoming ill-conditioned 2 The Background in a specified manner—either some elements of D are becoming infinite while the others are Θ(1), or else some are approaching zero while the remainder are Θ(1). In addition, Let us set the stage for Hamilton’s and Perelman’s work. techniques for sparse symmetric indefinite factorizations of matrices of the form 2.1 Homogeneous Geometries and Geometric Structures WAT 2 , (34) A homogeneous riemannian manifold (M,g) is one whose group of isometries acts tran- AD sitively on the manifold. Thus, the manifold looks the same metrically at any point. where D is diagonal and ill-conditioned as just described, are important. See, for example, n n Examples of homogeneous manifolds are the round sphere S , Euclidean space R ,and [11, 13, 40, 42]. hyperbolic space, Hn. A locally homogeneous manifold is one whose universal covering is homogeneous. Said another way, a locally homogeneous manifold is a complete riemannian 5NewProblems manifold such that given any two points x, y in M there are neighborhoods Ux of x and Uy of y andanisometryfromUx to Uy carrying x to y. We say that a locally homogeneous The flowering of interior methods, and in particular the realization that efficient algorithms manifold is modelled on a homogeneous manifold if every point of the locally homogeneous exist for a wide class of convex optimization problems, has led to the application of interior manifold has a neighborhood isometric to an open set in the homogeneous model. We are methods to a broad range of problems that were previously considered to be computationally primarily concerned with locally homogeneous manifolds of finite volume. intractable. 2 R2 H2 In dimension 2 there are four models for homogeneous geometries: S , , ,andG Certain problems involving eigenvalue optimization have been particularly amenable to R R∗ R∗ R where G is the group with the natural action of on . It turns out that there solution by interior methods; for details, see the excellent survey [22]. In the next section we are no finite volume locally homogeneous manifolds modelled on the fourth example, and summarize a few key ideas in semidefinite programming (SDP), an area of intense research hence we are concerned with only the first three models. The manifolds of the first type are during the past few years. the sphere and the projective plane, of the second type are the torus and the Klein bottle, and of the third type are complete hyperbolic surfaces of finite volume which can be either 5.1 The semidefinite programming problem compact or non-compact. This gives us the following well-known and classical theorem. Semidefinite programming may be viewed as a generalization of linear programming, where the variables are n × n symmetric matrices, denoted by X, rather than n-vectors. In SDP, Theorem 2.1.1. (Uniformization in dimension 2)LetX be a compact surface. Then we wish to minimize an affine function of a symmetric matrix X subject to linear constraints X admits a locally homogeneous metric modelled on one of the constant curvature models and semidefiniteness constraints, the latter requiring (in words) that “X must be positive above. The model will be positively curved if χ(X) > 0,flatifχ(X)=0, and negatively semidefinite”. This relation is typically written as X 0, a form that strongly resembles curved or hyperbolic if χ(X) < 0. inequality constraints in ordinary continuous optimization. (When X is a symmetric matrix, the condition X 0means“X is positive definite”.) n 2.2 Dimension 3 Let S denote the set of real n×n symmetric matrices, let C and {Ai} be real symmetric × In dimension three, up to isomorphism, there are 8 homogeneous geometries property for n n matrices, and let b be a real m-vector. The semidefinite programming problem is the following: which there are finite volume locally homogeneous examples. First, we have the constant (sectional) curvature examples: minimize trace(CX) (35) X∈Sn • 3 S of constant curvature +1. subject to trace(AiX)=bi,i=1,...,m (36) X 0. (37) 2 15 When the SDP problem is written in this form, its similarity to a standard-form LP (2) is hard to miss, but, not surprisingly, many extra complications arise in SDP. For example, the feasible region defined by (36) and (37) is not polyhedral, so there is no analogue of the simplex method. Furthermore, several major regularity assumptions are needed to obtain duality results analogous to those in LP. These assumptions will not be stated here; see [22] Perelman’s work on the classification of 3-manifolds for details. Nesterov and Nemirovskii [27] show that the function log det(X) is a self-concordant John W. Morgan barrier function for the semidefinite programming problem, which means that the SDP (35)– (37) can be solved in polynomial time via a sequence of barrier subproblems parameterized December 7, 2003 by µ:
minimize trace(CX) − µ log det X (38) X∈Sn 1 Introduction subject to trace(AiX)=bi,i=1,...,m. (39) Motivated by what was, by then, well-known for surfaces, Poincar´e formulated in 1905, Under suitable regularity assumptions, there is a unique sequence {Xµ,yµ},whereXµ a conjecture stating that a closed, simply connected three-manifold is diffeomorphic to is a symmetric positive definite matrix satisfying the constraints (36) and (37), and yµ is an S3. Developing tools to attack this problem formed the basis for much of the work in 3- m-vector, such that Xµ and yµ together satisfy the following “perturbed complementarity” dimensional topology. In the 1980’s Thurston studied 3-manifolds with riemannian metrics condition m of constant negative curvature −1, so-called hyperbolic manifolds. He formulated general X(C − yiAi)=µI, (40) conjectures about when a 3-manifold admits such a metric and proved various important i=1 special cases of his conjecture. This study led him to formulate a conjecture about the m with C − i=1 yiAi 0. Newton’s method cannot be applied directly to solve (36) and existence of homogeneous metrics for all manifolds, the so-called Geometrization Con- (40) because the matrix on the left-hand side of (40) is not symmetric. A primal approach, jecture for 3-manifolds. This conjecture includes the Poincar´e Conjecture as a very special first suggested in [1], is to replace (40) by the relation case. Thurston’s conjecture has two advantages over the Poincar´e Conjecture: X(C − yiAi)+(C − yiAi)X =2µI. • It applies to all 3-manifolds.
An analogous primal-dual method, called the “XZ + ZX method” for obvious reasons, is • It posits a close relationship between topology and geometry in dimension three. defined by finding (Xµ,yµ,Zµ), where Xµ 0andZµ 0, such that The conjectural existence of an especially nice metric on the three-manifold leads to a trace(AiX)=bi,Z= C − yiAi, and XZ + ZX =2µI. (41) more analytic approach to the problem of classifying 3-manifolds. Hamilton [H1] formal- ized one analytic approach by introducing the Ricci flow on the space of riemannian metrics Note the strong parallel between the two final equations in (41) and the primal-dual equa- (on manifolds of any dimension). He then conjectured that starting with any metric on tions (28) in linear programming. a compact three-manifold, the Ricci flow should produce a one-parameter family of met- Semidefinite programming is an extremely lively research area today, producing new rics converging to the nice metric as postulated by Thurston’s Geometrization Conjecture. theory, algorithms, and implementations; see the surveys [33] and [34]. There are many technical issues with this program – for example, one knows that in general the Ricci flow will develop singularities in finite time. Thus, a method for analyzing these 5.2 New applications of interior methods singularities and continuing the flow past them must be found. Furthermore, even if the flow goes on for all time, there are many complicated issues about the nature of the limit Interior methods are playing major roles in at least two areas: approximation techniques at time t =+∞. for NP-hard combinatorial problems, and system and control theory. Hamilton [H1, H2, H3, H4, H5] made much progress on this program and established In the former, it has recently been shown that certain semidefinite programs and NP- many crucial analytic estimates for the evolving metric and curvatures. He also showed hard problems are closely related in the following way: solution of the semidefinite program that in special cases the Ricci flow could indeed be used to establish the Geometrization leads to an approximation whose objective value is provably within a known factor of the Conjecture. More recently, building on this work, G. Perelman, in a series of three preprints optimal objective value for the associated NP-hard problem. For example, a semidefinite [P2, P3, P4] has claimed to surmount all of the various technical and analytic difficulties to program formulation leads to an approximate solution of the max-cut problem whose objec- tive value is within a factor of 1.14 of the optimal value; see [18]. This kind of relationship 1 16 guarantees that good approximate solutions to NP-hard problems can be computed in poly- nomial time. Interior methods are important in system and control theory because of their connection with linear matrix inequalities, which have the forms p p F0 + xiFi 0orF0 + xiFi 0, (42) i=1 i=1 where x is a p-vector and {Fi} are real symmetric matrices. Many constraints in system and control theory, including convex quadratic inequalities, matrix norm inequalities, and Lyapunov matrix inequalities, can be expressed as linear matrix inequalities. It is straight- forward to see that the forms (42) allow the variables to be symmetric matrices. Numerous problems in system and control theory involve optimization of convex func- tions of matrix arguments subject to linear matrix inequalities. Because these are convex programming problems, it is possible to apply polynomial-time interior methods. For de- tails, the reader should consult [3].
6 Summary
The interior point revolution has had many highly positive results, including • a deeper and more unified understanding of constrained optimization problems;
• continuing improvements to theory and methods;
• more algorithmic options for familiar problems, even for linear programming;
• the ability to solve new problems. One could argue, however, not entirely whimsically, that the interior-point revolution has had some negative consequences. For example, both teaching linear programming and solving linear programs are much more complicated than they used to be. With respect to the former, instructors in linear programming courses face increased pedagogical challenges. Before 1984, it was perfectly acceptable simply to describe the simplex method; today, any credible treatment of linear programming needs to include interior methods. Similarly, someone with an LP to solve can no longer be content with mindless application of the simplex method. On balance, the interior revolution has energized and expanded the field of constrained optimization. Although the revolutionary pace has (inevitably) slowed down since its first heady days, ample opportunities remain for many further years of lively and innovative research.
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æe×ÒäÝ åçÛµ<¡Ý¡ þ.éeá¢æeïçï ¢ L L L Ó Ó ¡G "ã}ã}4(Éì :s<££ç7JO354=7ªOA35,W3E02I DnF,h0]63Ë@6ö ÷¡#Sï;éeÚm: ' ~¢Ý#Tï;éeÚb: ' +M,¨N!OP0RQ5OÉ7JO3k+M,WVP38,6+ ø :f]6+^V`@º02InXZ#SïçéeÚ: '\[ a .My4bC/| "~nlnrg t zj~n n¤Gbm2rGpSmb~Spn| I¥FSm6bn| rVrBopn|v{mI~y8r¢p$Alq| ¢lXÌIXÄÍQÐÈ<| ~Qt ~Sr «QrV wpS ~Í¢ÉQÎ pSmb~n t G 4~ 4GnlnmC|v{m |vGv2mIz}|FSmwt §4oqout| mIzúroprnmXpmI~n t FsVrg n«mbpCv<4 ~©§F~S ` k"lmIr¢pSmwvÌYClm6 Q 4n4F «mbpnvG4 ~¯§F~Sk"lnmbrQpm6v wQsgm> ~Smbz¦rVnlnm¥oprQsut mwvÆlnmbpSmw 20 P Î ¦ Ó ¦ " £m £ L ¡G "ã}ã} (ì wK~;âgöÉf7JO354$wâgöÉ/a L ,W+^+ aE#þ¦â6áCâ îMâ îMéeá5£íåç× ÜCàÒâ áCå;×ÒÚ ÷ u »#z7~;Ä{#¥u{'5'5' ÖFþqm#¥ub'qöãÆ ÜCàÒâ6×äm#¥u ¡'±Àm#¥u{'Q¡çö£Æ ø ¢ Æ å;Û<æÚeáCéeÙÒë àÒâ6áCâ6þ.éeáCâ##5#¡u ¡§'Qå¤'qo#¡u ¡'QåXæeÛ#âËöÉ÷æe×ÒäãÛbé ÛCåç×Òè6â WHAT IS MOTIVIC MEASURE? ¢¨ ¡¤å ¡ å ¡ å ¡ å ¡jå m#¥u{' #zm#¥u{' ' om#¥u ' o#y#¥u ' 'qom#¥u 'hõ THOMAS C. HALES L ¡G "ã}ã}=à(ì wK~;âgöÉpDM4V¨w¨z%â êéuäe@É7JO354çwqz%â êéuäÓÌæÆ|Ì¡a L ÷ u îMâ<àÒæ eâ¢ÜCàÒæeܳènfúäÒå¥ §å;äÒâ6Û-#zè{'Km#zf' àÒâ6áCâ6þ.éeáCâGu C¢äÒå¡ uåçäÒâ6Û ,W+^+ aªéeá¥æeקóçm#¥ub'ö ø ¢b¨ ¡gFwå ¡ å ¡ å u îeßî¢àÒåçèà äÒå¥ uåçäÒâ6Û`u u ßî<àÒå;èà äÒå¥ §å;äÒâ6Û #¡u '¯ém#¥u ' l>âäÒÙÒè6å;×ÒÚÃêé§ä$7"ßÛCéãÜCàÒæeÜ ¢ Abstract. This article gives an exposition of the theory of arithmetic ¡ ¡ å å Ä{#¥ub'äÒå¥ §å;äÒâ6Ûou eßîMâäÒâäÒÙÒè6âÄÜCàÒæeÜx#¡ub' Ím#¥u 'ã#¡u ' ê#¡ub' åç× ÷ u »#z7~;Ä{#¥u{'5'jþ.éeá ø motivic measure, as developed by J. Denef and F. Loeser. ÿ æeï;ï36öéÆ ßªæe×ÒäÛCém#¥u{'Q¡gFwå!ü þ.éeáXæeï;ïw*ö9Æ ð¦éqîãÆ åçÛXæÄè6ó§è6ï;åçèÚeáCéeÙÒë$ß(ÛCéúÜCæ&£uåç×ÒÚ+ÜCé âæ ¢ Úeâ6×Òâ6ábæeÜCéeá¢éeþIÆ îâ äÒâ6äÒÙÒè6â ÜCàÒæeÜ¢ÌÎÆ|ÌäÒå¥ §å;äÒâ6Û`wµëâ ¢ ÿ ÿ ÿ $â6Ü^ì âBÜCàÒâBÛCÙ ÚeábéeÙÒëéeþm# »W@ '¤í¥Úeâ6×ÒâáCæeÜCâ6ä ó/:æe×Òä7 õÒå;×Òè6â!:å;Û2×ÒéeÜæë©éqîâ6á2éeþ¾7"ßVÜCàÒâ Ý 1. Preliminary Concepts ¢ å;×ÜCâ6ÚeâáCÛ;:-î¿7 î<å;ÜCàï¤~Q¨<¡Þ æeáCâXäÒå;ÛbÜCå;×Òè6Ü àÒâ6áCâ¦æeábâYXðÌæì)ÌÛCÙÒèà åçקÜCâÚeâ6áCÛMî<å;ÜCàúÞSrï¤~Qqr ÌÎìÌ ¢¨ ÿ âè6ée×ÒÚeáCÙÒâ6×Ü #TêéuäU@"']ß}ÛCæó æe×ÒäÛCéÜFîMéê ÙÒÛCÜ There is much that is odd about motivic measure if it is judged by measure theory in the sense of twentieth century analysis. It does not fit neatly with î 7 zs:-ñ@7óò #Sêé§äk@¡'hõ : the tradition of measure in the style of Hausdorff, Haar, and Lebesgue. It is best to view motivic measure as something new and different, and to ÿ ÿ ÿ ÍGóUÝ$â6êêæ XÜCàÒâ6ÛCâ¦åçקÜCâÚeâ6áCÛBæeáCâ éeÜCàúå;×mÉ Í<ódÝâêêæÝ ÜCàÒâå;áBäÒåzô©â6áCâ6×ÒèâåçÛMäÒå¥ §å;ÛCå ï;â óÈÌÎÆ|Ì8ß ¢ æe×ÒäÜCàÒâáCâ6þ.éeáCâ recognize that when it comes to motivic measure, the term ‘measure’ is used loosely. [ î  õY  õY rs: õ 5¥&' ÌÎÆ|Ì rðÌ : 7 $:-ñj7óòÌ rZ#¥:ó7' # Motivic measure will be easier to understand, once two of its peculiarities are explained. The first peculiarity is that the measure is not real-valued. à(ªà()¡G§ K e! ädãÌæÆ|Ì ( âî<å;ÛCà ÜCéÛCàÒéîùÜCàÒæeÜGÜCàÒâ6ábâæeáCâêæe×ó äÒå;ÛCÜbå;×Òè6Ü<â6ï;âêâ6×ÜCÛ<éeþIÆ ÖFþ;ó #¥u{'h~Qm#¥u{'!ö u î¢åçÜCà ÷ Rather, it takes values in a scissor group. An introductory section on scissor ø Ì ¢ £m#¥u{'*#Têéuä=#z7~;Ä{#¥ub'y'5'jÜCàÒâ6×íîMâè6æe×î<áCåçÜCâeó #¡ub'!9#¡ub'µzöÄb#¡ub'@rH#¡ub'ùêéuä7 þ.éeá ÛCéeêâ ó #¥u{'Yz groups for polygons will recall the basic facts about these groups. The ÿ ë¨éeïçóu×Òéeêå;æeïIrH#¡ub'/ö ÷ u à§ÙÒÛjåçþGóæe×Òä éeÜCàÃàÒæ eâÛCêæeï;ïçâ6á¦äÒâ6ÚeáCâ6âÜCàÒæe×ÈÄ ÜbàÒâ6×÷rH#¡ub'/z Þ ø second peculiarity is that rather than a boolean algebra of measurable sets, ¢`¨ #Sêéuä7'Gæe×Òä ÛCé ó #¥u{'z#¡ub'*#Sêé§ä7' à§ÙÒÛ¢æeï;ï&ë©éeï;óu×Òéeêå;æeïçÛBéeþÜCàÒâþ.éeáCê #¥ue wP'QÇ ô© © ¢ ¨  ¡ we work directly with the underlying boolean formulas that define the sets. ÿ éeþ¨äÒâÚeáCâ6âçs #SÜCàÒâ<äÒâ6ÚeáCââ>éeþ Ä{#¥u{'5'Aæeábâ¢äÒåçÛCÜCå;×ÒèÜAâ6ïçâ6êâקÜCÛéeþ Æßeæe×ÒäÄÜCàÒåçÛAÚeå¡ eâ6Ûæï;éîMâ6á éeÙÒ×Òä ¶ The reasons for working directly with boolean formulas will be described in þ.éeá+Æ æ×Òâè6æe×íÛCàÒéî ÜCàÒæeÜ åçÛÜbàÒâéeáCäÒâá éeþ>7#Sêé§äk@"'æe×ÒäÛbéãå;þée×Òâè6æe×ÛCàÒéqî ÜbàÒæeÜÜCàÒåçÛ ¶ ¢ ÿ æeï;ÙÒâå;Û>ïçæeáCÚeâjÜCàÒâ6×îâ è6æe×ÚeâÜ¢Úeé§éuä§ï;éîMâ6á éeÙÒ×ÒäÒÛ>ée×$Æ a second introductory section. ¢ !¦ÒáCÛCÜAëÒáCâ6ëÒáCåçקÜwßæe×ÒäÜbé ëÒáCé§ eâ¢ÛbÙÒèQà àÒå;ÛAîMæeÛAî<àÒæeܥئÚeáCæqîæeï4ßgòæóeæeï(æe×ÒäÄõÒæeöuâ×Òæ äÒå;äå;×ÜCàÒâ6å;á After these two introductory remarks, we will describe ‘motivic counting’ ¨ ÿ @úâ6öuåçÛCÜGÜCàÒâ6ó ×Òâ6â6äÒâäÜCéÙÒÛbâ äÒâ6â6ëÜbéuéeï;ÛGéeþæe×Òæeï;óuÜbå;èjקÙÒê â6á>ÜbàÒâ6éeáCó Ö× ÜCàÒâ6åçá¢ÛCâè6ée×ÒäëÒáCâ6ëÒábå;×Ü]ß ¢ in Section 2. Motivic counting is to ordinary counting what motivic measure ÿ ÿ ÿ å;×ÒÛbëÒå;áCâ6ä óãæ áCâ6êæeá5£ãéeþ;Ý$â6×ÒÛCÜCáCæªß¨ÜbàÒâ6óãîMâáCâ æ ï;â ÜCéúábâ6ëÒï;æeè6â óÓÌæì)Ì}å;×ÜbàÒå;Û¦áCâ6ÛbÙÒï;Ü]ß©î<àÒå;èà ¶ is to ordinary measure. Motivic counting will lead into motivic measure. æeï;ïçéqî<ÛjÜCàÒâ6ê ÜCé§Úeå¥ eâæe×âקÜCåçáCâ6ïçó âï;â6êâ6קÜbæeáCóëÒáCéuéeþAéeþÜCàÒâå;áÜCàÒâ6éeáCâ6êß"æe×Òä ÜCé§Úeâ6ÜjæãÛCÜCábée×ÒÚeâ6á áCâ6ÛbÙÒï;Ü>î<àÒâ6×ÜbàÒâ6ó äÒéå;ס eé&£eâÜCàÒâ äÒâ6âë¨â6áXâÛCÜCå;êæeÜCâ6Û ¢ 1.1. Scissor Groups for polygons. Motivic volume is defined by a process ¡G "ã}ã} ÿ(mø_^FPF+MI3S7JODM7oó #¡ub'h~Qm#¥u{'oö ÷ u N;027JOºó #¥u{'µz%#¡ub'*#Sêé§ä}#z7~;Ä{#¥u{'5'5'h>µD-4mVkó~Qçö ø that is similar to the scissor-group construction of the area of polygons in L eagì ófDM4mV# B¤+M7JO OD-1"3qVP39.§,W3¤3#sðÌæì)Ì7ªOA354çó #¥u{'¯zëm#¥u{'6#Sêé§ä7'a Æ the plane. To draw out the similarities, let us recall the construction. It L ,W+^+ aËÝ$â6Üxù#zú'(Jæ ó #zú'mä#ú' ÖFþqrdöÉ÷ÜCàÒâ6× ¢ determines the area of a polygons without taking limits. Any polygon in the plane can be cut into finitely many triangles that t t t t t #¥u 'I ó #¥u 'mÈ#¡u 'Gzó #¡ub' Èm#¥u{' z¡Þ #Sêéuä}#7~;Äb#¥u{'5'5'¤õ ù can be reassembled into a rectangle of unit width. Figure 1 illustrates three steps (2, 3, and 4) of the general algorithm. The algorithm consists of 5 elementary transformations. (1) Triangulate the polygon. (2) Transform triangles into rectangles. (3) Fold long rectangles in half. (4) Rescale each rectangle to give it an edge of unit width. (5) Stack all the unit width rectangles end to end. The length of the unit width rectangle is the area. An abelian group encodes these cut and paste operations. Let F be the free abelian group on the set of polygons in the plane. We impose two families of relations: 1 2 THOMAS C. HALES m .kü.Õ ¦MÕ©¨ k k k ¦ Ó k Ó .Õ £ £ £ ÿ ábå;ÜCåç×ÒÚ : å;× å;×ÒæeábóßÛCæóæeÛµ: ÝB¡ v ÝB¡ v£¢;¢;¢Pv ÝB¡G¤î<å;ÜCà wKôUX w X¥¢;¢;¢X w¦¦¨< Þªß C A [ Ì ï;âÜô#¡ub' ó #¥u{' æe×Òä ÜCàÒâ6×K #¡ub'¨z§K ô#¥u{'hó #¥u{'#Têéuä=#¥:!~yu ZW'5' þ.éeá0$ ^~¢Ý"~-õ-õ-õ¤~Q¨ ¡ F ¡ ¢ A àÒâ6áCâ6þ.éeáCâ ¨ ª«ª«ª¬ ¤ C [©© x [©© [©© #¥uUvTw&' #Têéuä=#¥:!~yu W'5'hõ K¦-#¥u{'GzZ#¥uUvfw&' A à§ÙÒÛ<îMâ¦àÒæ§ eâèéeêëÒÙÒÜCâ6ä=#¥uSvswP'yx#Sêé§ä}#¥:!~yu W'5'Bå;×*wôHv¨Yr¥AïçéeÚm: ÛCÙÒèàãÛbÜCâ6ëÒÛ]ßuî<àÒâ6áCâjæ ¨ ÛCÜCâë å;ס eéeï¥ eâ6Û$ê ÙÒï;ÜCåçëÒï;ó§å;×ÒÚ>ÜFîMé>ë¨éeï;ó§×Òéeêåçæeï;Û$éeþ}äÒâ6ÚeáCâ6âss@î<å;ÜCàè6éuâ¤è6åçâ6קÜbÛ{å;×¢nqÞ"~ ^~-õ-õ-õ¤~y:ËeGp}ß æe×ÒäáCâäÒÙÒè6å;×ÒÚ #Sêé§ä}#¥:!~yuW'5' ¢ #²³²-´µ²-´$¶¸·"%¹i·"%\²#&%\º ¹i·"% »¼÷*"-¾½-²#&,¿¶¸· º ® ¯±° ¯ ÖF×úÜCàÒâXÛbâ6è6ée×Òä àÒæeï;þ&éeþÜbàÒâXÜCæe北 îMâXî<å;ïçï(ëÒáCé eâ>ÜCàÒâXÜCàÒâ6éeábâ6ê éeþ&ئÚeáCæîMæeï ß/òæqóeæeïªæe×ÒäúõÒæeöuâ×Òæ ¢ âjî<å;ï;ï&æeÛbÛCÙÒêâÜCàÒæeÜBîMâæeáCâjÚeå¥ eâ6קæeקéuäÒäãåçקÜbâ6Úeâ6á!:ôî<àÒå;èà îMâE£§×Òéqî å;ÛG×ÒéeÜGæ ë©â6áCþ.â6èÜ>ë¨éîMâ6áwß Ì [ æe×ÒäÃàÒæeÛj×Òé§ëÒábå;êâ þ.æeèÜCéeáEr @ ªéeáXÛCåçêëÒï;åçè6å;ÜFóîâ î¢åçï;ïæeÛbÛCÙÒêâÜCàÒæeÜË:ÆàÒæeÛjéeáCäÒâ6áEX ñ#Tï;éeÚm: ' ¢ êé§äÒÙÒï;é©@ß}æe×ÒäãÜCàÒæeÜ x x #5W' #¥udvTw&' zsu vfw #Têéuä=#¥:!~yu W'5' þ.éeáâ6æeèàÆå;×ÜCâ6Úeâákw"~ r w r à î<àÒâ6áCâ§îMâ ÜCæ&£eâ*ÃÀ¥ @ï;éeÚ: #ªæ×Òâ èæe×üáCâ6ëÒïçæeè6â§â6æeèàüéeþ ¢ ÿ ÜCàÒâè6ée×ÒÛCÜCæe×ÜCÛ ø¥ñeû æe×ÒäøÁ¥eû ó øWûãî<å;ÜCàíâöuÜCáCæ îMéeá5£"' ÍGó àÒâ6éeáCâ6ê îMâd£u×Òéî ÜCàÒæeÜÜbàÒâ6ÛCâ ¢ ¨ àóuë¨éeÜbàÒâ6ÛCâ6ÛXàÒéeïçäãåçþH:÷åçÛ¢ëÒábå;êâeß(ÛCéîMâ ê ÙÒÛbÜ>ÛCàÒéîùÜCàÒæeÜ<ÜCàÒâ6ó è6æe×Ò×ÒéeÜ>àÒéeïçäãåçþ:÷åçÛ¢è6éeêë¨éeÛCåçÜCâ ¢ ÿ ÿ Ý$â6ܸ7 âæ¢ëÒáCå;êâäÒå¡ uå;äÒåç×ÒÚG: âè6æe×þ.æeèÜCéeáAu&kAå;×ÜCé>åçáCáCâ6äÒÙÒèå ï;âÛ{å;× ÷ u ßæeÛ #¡ub']ß ø ¢mÌ Ã ·G î<àÒâ6áCâ #¥u{'>å;Û>ÜCàÒân@§ÜCàãè6óuè6ïçéeÜCéeêåçè ë©éeï;óu×Òéeêå;æeï ß(î<àÒéeÛCâ áCé§éeÜCÛ>æeáCâ ÜCàÒâëÒáCåçêå;Übå¥ eâµ@ÜCàáCé§éeÜCÛXéeþ à ÿ ÿ ÙÒ×Òå;ÜFó ÝâÜÄ{#¥u{' â æe×ãåçáCáCâ6äÒÙÒèå ï;âþ.æeè6ÜCéeá¢éeþ #¥u{'#Sêéuä7' àÒâ6×ï#5W'GåçêëÒï;åçâ6Û<ÜCàÒæeÜ ¢ ¢;¨ à x x SÝ&' #¥uUvfw&' zsu vTw #Têéuä=#z7~;Ä{#¥ub'y'5' Figure 1. Triangles transform into unit width rectangles # `wK~ orw¨rÃß}ÛCå;×Òè6âk#z7~;Ä{#¥ub'y'<äÒå¡ uåçäÒâ6ÛS#¡:!~yu{W' by scissor and congruence relations. Later, we will transform þ.éeá<â6æeèàå;×ÜCâ6Úeâ6á ¢ #Sêé§ä}#z7~;Ä{#¥u{'5'5'&æeáCâ áCâ6æeïçï;óXÜbàÒâAâ6ïçâ6êâקÜCÛ$éeþuÜbàÒâ¥ábå;×ÒÚ ÷ u »#z7~;Ä{#¥ub'y']ß ring formulas into algebraic varieties by scissor and congru- àÒâAèée×ÒÚeáCÙÒâ6×Òè6âAè6ï;æeÛCÛbâ6Û ø ¨ î<àÒå;èà åçÛMå;ÛbéeêéeáCëÒàÒåçè¦ÜCé ÜCàÒⵦÒâ6ïçä éeþw7 Å â6ï;âêâ6×ÜCÛE#Sî<àÒâ6áCâ åçÛMÜCàÒâ¦äÒâ6ÚeáCââéeþ-Ä' ÖFקëÒæeábÜCå;è6ÙÒïçæeá ence relations. ¶ ¢ Å ÿ ÜCàÒâ×Òée×Òýª 6â6áCéúâï;â6êâ6קÜbÛ¦þ.éeáCê æúèóuè6ïçå;è ÚeábéeÙÒëéeþ{éeáCäÒâ6á!7 #Í<âï;éîùå;Ü>î<å;ïçï âé§è6è6æeÛCåçée×Òæeï;ï;ó ¢ è6éeס eâ6×Òå;âקÜXÜbéÛCÙÒëÒëÒábâ6ÛCÛ¦ÜCàÒâøêé§äÒû×ÒéeÜCæeÜCåçée× ' Scissor relations. If P is a polygon that can be cut into polygons P and ¢ ÿ ÿ ÿ $â6Ü)Æ â ÜCàÒâ$#Sè6ó§è6ï;åçèM' ÛCÙ ÚeáCéeÙÒë÷Úeâ6×Òâ6áCæeÜbâ6ä óºuÉv ^~yu*vùÝ"~-õ-õ-õh~yu*v à ð¦éeÜCå;è6âúÜbàÒæeÜ å;þ 1 Ý ¢ m#¥ub'Y #¥uUv|wP'QÇ öÈÆ ÜbàÒâ6× © P2, then ô © ¡  [P ] = [P1] + [P2] x Ç x Ç x x #5#¡uUvfwP' ' z #¥u vTw&' om#¥u ' #Têéuä=#z7~;Ä{#¥ub'y'5'hõ m#¥u{' © Congruence relations. If P and P 0 are congruent polygons then © ¡ ¡ 0 ÿ âäÒâM¦Ò×Òâ)ÉÜbé â ÜCàÒâÛCâ6ܦéeþ¥å;×ÜCâ6Úeâ6ábÛ+rãþ.éeá¦î<àÒå;èà #¥u '2m#¥u{' þ.éeájæeï;ïºöÊÆ ð¦éeÜCâ ÜCàÒæeÜ t [P ] = [P ]. t Ì ¢ 7~y:CöÉ ¢ ÿ ÿ ÙÒá ëÒï;æe×å;Û{ÜbéÚeå¡ eâMÙÒëÒë¨âá¥æe×ÒäïçéqîMâá éeÙÒ×ÒäÒÛ ée× ÜCàÒâBÛCå¡ 6âGéeþ3Æ ÜCéjâ6ÛCÜCæ ïçå;ÛCàæ¦è6ée×ÜCáCæeäÒåçè6ÜCåçée× The scissor group Spoly of polygons is defined as the free abelian group æ subject to these two families of relations. In some sense, this entire article ¢ à( ()Ë6â ⯠" e¯ ädÍÌÎÆ|Ì ( is an exploration of scissor and congruence relations in diverse contexts. By Ï SmÐgÌÑuÐTÓÒgÌÔÑ}ÐGÌv{r6z ÌÔV¢ÕQÖVÌÔÑ}ÐSÐÐAlnmbpm#V | ~QÄ|bSmbQmbpjQnzÐgÌÑuÐ×Õ¿ÒgÌÑuÐ×ÕÁÖVÌÑuÐصÙ-Ú ÑNÛ | ynlnmbpm and by, we will construct several closely related scissor groups Spoly, Scount, mX"pn| mb| ~~ ÜÒÌÔÑ}ÐÕÞÝeÌÔÑ}ÐwØ0Ù-Ú ÑßÛ/y8r¢p$Alq| ¢lÐgÌÔÑ}оXÒVÌÑ}Ðwà)VÜÒÌÔÑ}ÐKáDÖVÌÔÑ}ÐÝgÌÑ}Ð Sring, Scover, and Smot, each constructed as a free abelian group modulo scissor and congruence relations. Theorem 1.1. The polygon scissor group Spoly of polygons is isomorphic to the additive group of real numbers R. Under this isomorphism, the real number attached to the class [P ] of a polygon is its area. Proof. A group homomorphism from Spoly to R sends each class [P ] to its area. It is onto, because there are polygons of every positive real area, and negations of polygons of every negative real area. By scissor and congruence WHAT IS MOTIVIC MEASURE? 3 P ¦ Ó ¦ " £m £ relations, every element of the scissor group is represented by the difference ÜCàÒæeÜ<æÚeå¡ eâ6×ãëÒáCåçêâå;Û<ëÒáCå;êâ/ß}æe×ÒäëÒáCé§ eâ6Û>ÜCàÒæeÜ<ëÒáCåçêæeï;åçÜFóúÜCâ6ÛCÜCåç×ÒÚåçÛ¢åç×ãè6éeêëÒï;â6ö§å;ÜFó è6ï;æeÛCÛ!¡Ê ¢ of two unit width rectangles. To be in the kernel, the two rectangles must (5(¥ T T! K ª ª ( have the same area; but then they are congruent, and their difference is zero àÒâ ×Òâ6î îMéeá5£ãéeþ2ئÚeábæqîMæeï ß(òæóeæeïæe×ÒäÃõÒæeöuâ6×ÒæúåçÛXê ÙÒèàÃÛCå;êëÒï;â6áXÜbàÒæe×êæe×óãéeþ ÜCàÒâêéeáCâ ¨ element of Spoly. Thus, the homomorphism is also one-to-one. ¤ ÿ áCâ6èâ6קÜjäÒâM eâï;éeëÒêâקÜCÛjåç× ÜbàÒå;ÛXÛCÙ ñâ6èÜ àÒâ6åçá¦ÛCÜCæeáCÜbå;×ÒÚúë¨éeåçקܦåçÛXÜCàÒâ þ.éeïçï;éî¢åç×ÒÚúáCâ6ÛCÙÒïçÜ]ߨî¢àÒåçèà åçÛ ¢`¨ ÿ Úeéué§äãâ6ö§â6áCè6åçÛCâþ.éeá>æe×ãâï;â6êâ6קÜbæeáCó ×ÙÒê âáXÜCàÒâ6éeáCó è6éeÙÒábÛCâ ¢ The area function on the set {P } of polygons thus factors through Spoly. L L ¡¨ê " § Kã (ÉìR47839."38,g:T02IgF,h02]*3`0 DM4V +M4?jik0 #¡udvîW'yxezsu{xYvî=#TêéuäU: 'Y024 ÷ u a ø {P } → Spoly → R (1) x L ,W+^+ aGõÒåç×Òè6âï#¥uºv W'yx6 #¥uxkvüW' u ßAîâàÒæ eâãÜCàÒæeÜ©uxevüçz #¥uvüW'yx ô©wj ô P 7→ [P ] 7→ area(P ) xaF x þ.éeá¢æeïçï\å;× ÜCàÒâábæe×ÒÚeâkor¼rs:6T ÖþA:o7§å;Û<ëÒáCå;êâ #Sêéuäk: 'Gåçþæe×Òäãée×Òïçóúå;þA:ôäÒå¡ uå;äÒâÛ ¢ We might ponder which of these two maps (P 7→ [P ] or [P ] 7→ area(P )) E ÿ ÜCàÒâ6×+7æeëÒë¨âæeáCÛåç×ÜCàÒâ<קÙÒêâ6áCæeÜCéeáAéeþ ÙÒÜAå;Û¥ï;æeáCÚeâ6á ÜCàÒæe×$ßeæe×ÒäÄÛCéäÒéuâ6Û¥×ÒéeÜ¥äÒå¥ uåçäÒâ/ßeæeקóÜCâ6ábê captures the greater part of the area-taking process. Motivic measure com- å;× ÜCàÒâ äÒâ×Òéeêå;×ÒæeÜbéeá ¢ x ÿ !: å;Ûèéeêë¨éeÛbå;ÜCâÄïçâ6Ü7 âÄæëÒáCå;êâÄäÒå¥ §å;äÒåç×ÒÚ : ÖF× ÜCàÒâúâ6ö§ëÒæe×ÒÛCå;ée× :Ë#¥:$pW'^#¡:C ÖFþ mits to a position on this issue: the first stage (P 7→ [P ]) is identified as the ¢ E &'Aõ-õ-õ§#¡: f#7nfW'y'h»¤7{í§îMâjÛCâ6âÜCàÒæeÜBÜCàÒâée×Òï;óÜCâ6áCêÛ^7 äÒå¥ §å;äÒâ6ÛGæeáCâjÜCàÒâ: å;קÜCàÒâjקÙÒêâ6áCæeÜCéeáGæe×Òä Ý area-taking process and the second stage [P ] 7→ area(P ) is a specialization ô F 7úå;× ÜCàÒâ¦äÒâ×Òéeêå;×ÒæeÜbéeá]ߧæe×Òä ÛCéå;þ\7 tåçÛBÜCàÒâ¦ï;æeáCÚeâÛCÜMë¨éîMâ6áBéeþw7úäÒå¥ §å;äÒåç×ÒÚn: ÜbàÒâ6×¢7 t åçÛBÜCàÒâ ÜCàÒâ of the area. In this case, specialization is an isomorphism. Our approach to x x ï;æeábÚeâ6ÛCÜ<ë¨éîMâ6á>éeþ7ãäÒå¥ §å;äÒåç×ÒÚ Ä}æe×ÒäãÜCàÒâáCâ6þ.éeáCâµ:ôäÒéuâ6Û<×ÒéeÜ>äÒå¡ uåçäÒâ ¢ measure in this article is decidedly unsophisticated: taking the measure of E E ÿ å;ÛÜCàÒâ æeÛCå;Û éeþ¢ÜbàÒâ ×Òâ6î ëÒáCå;êæeï;å;ÜFóíÜbâ6ÛCÜGJ à§ó äÒée× : Ü îâ è6éeêëÒÙÒÜCâ àÒå;Û ÛCå;êëÒï;â ÜCàÒâ6éeábâ6ê something consists in mapping that thing into its scissor group, P 7→ [P ]. ¨ Ì x x #¥uµv W' ï#¡u v W'#Sêéuäk: 'Aæe×ÒäÄäÒâ6ÜCâáCêåç×ÒâXî¢àÒâÜCàÒâ6áMéeáA×ÒéeÜ:äÒå¡ uåçäÒâ6Ûâ6æeèàúè6é§âèå;â6×Ü5å àÒåçÛ ¨ ÿ ÿ É æeï;å;äëÒáCå;êæeï;å;ÜFóÃÜCâ6ÛbÜ]ß ÙÒÜè6éeêëÒÙÒÜCå;×ÒÚ}#¥u6v W'Rx #Sêéuäk: 'å;Ûé §å;éeÙÒÛCïçó ÛCïçéqî ÛCåç×Òè6âÄåçÜ î<å;ï;ï å;Ûæ 1.2. The measure of a formula. Traditionally, we take the measure of a ¡ eéeï¥ eâÛCÜCéeábå;×ÒÚS:ôè6éuâ¤è6åçâ6קÜbÛ^í å;× set X = {X | φ(x)} (say a subset of a locally compact space), but we do not è6ÙÒïçÜFó§åçÛ>ÜCàÒæeÜ<ÜCàÒâæe×ÒÛCîMâ6á¢àÒâ6áCâåç×" eéeï¡ eâ6Û¢êæeקó§è6é§âè6åçâ6×ÜCÛ¨#SæeÛ¢ÜCàÒâäÒâ6ÚeáCâ6âåçÛ õÒåç×Òè6âéeÙÒá>äÒå take the measure of the formula φ defining a set. With motivic measure, we t']ߪée×Òâ åçäÒâ6æúå;ÛXÜbéÄè6éeêëÒÙÒÜbâ êéuäÛbéeêâÛCêæeï;ïäÒâ6ÚeáCâ6â ë©éeï;óu×Òéeêå;æeï"æeÛXîMâï;ïæeÛ>êéuä :Gß(ÛCé ÛCéúàÒåçÚeà take the measure of the formula directly. Concretely, the formula ÜCàÒæeÜG×Òâ6åçÜCàÒâ6áGÜCàÒâè6éuâè6å;âקÜCÛ<×ÒéeáGÜCàÒâjäÒâ6ÚeáCâ6âÚeâ6ÜGï;æeábÚeâ àÒâjÛCåçêëÒï;âÛCÜ¢ë©éeï;ó§×Òéeêå;æeï©éeþäÒâÚeáCâ6âo@ ¢ ¨ gugT õÒéÄîâè6éeÙÒïçäÉ eâ6áCåçþ.ó î¢àÒâ6ÜbàÒâ6á å;Û<ë¨âáCàÒæeëÒÛ 2 2 ¢ (2) ‘x + y = 1’ x x z u vî #Sêé§ä}#¥:!~yu W'5'hõ #¥udvîW' defines the circle 2 2 ÿ â èéeêëÒÙÒÜCâ6ä÷áCæeëÒå;äÒïçóß æe×Òäôå;ÜåçÛ ÜbáCÙÒâ þ.éeá æe×ó ëÒáCå;êâd:ü#SæeÛæ èée×ÒÛCâM¤uÙÒâ×Òè6âãéeþ¢ÜCàÒâ àÒå;Û è6æe× (3) {(x, y) | x + y = 1}. ¨ ÿ ÿ æ é eâM'wß ÙÒÜXåçÜ>å;Û¢ÙÒ×Òè6ï;âæeá¦î¢àÒâ6ÜbàÒâ6áXÜCàÒåçÛXþ.æeå;ïçÛXþ.éeáXæeï;ï"è6éeêë¨éeÛCåçÜCâµ:Ææe×ÒäÜCàÙÒÛXëÒáCé§ uåçäÒâ6Û ÜCàÒâ6éeábâ6ê With motivic measure, we take the measure of the equation of the circle æ ëÒáCåçêæeï;åçÜFóüÜCâ6ÛbÜ àÒâêæeå;×xÜbàÒâ6éeáCâ6ê #TæeÜÄÜCàÒâÛCÜCæeábÜÄéeþjÜCàÒâÜCæe北"'ëÒáCé§ uåçäÒâ6Ûúæ êé§äÒ奦Òè6æeÜbå;ée× ¢T¨ ÿ âÛCàÒéî<× ÜbéíÛCÙÒè6è6ââ6ä þ.éeáÄëÒábå;êâÛúæe×Òä þ.æeåçï>þ.éeáÄèéeêë¨éeÛbå;ÜCâ6Ûwß éeþ¦ÜCàÒåçÛúè6ée×ÒÚeáCÙÒâ6×Òè6âeßGî<àÒå;èàè6æe× (Equation 2) rather than the measure of the circle itself (Equation 3). At- § uåçäÒå;×ÒÚ§æ ë©éeï;ó§×Òéeêå;æeï2Übå;êâëÒáCåçêæeïçå;ÜFóÜCâ6ÛbÜ Ö× ÜCàÒâ ÛCâ6è6ée×Òä ëÒæeáCÜéeþAéeÙÒáÜbæe北îMâ ÛCàÒæeïçï ÜCàÙÒÛ ëÒáCé tention shifts from sets to formulas. ¢ å;ס eâ6ÛCÜbå;ÚeæeÜCâÜCàÒå;Û<å;× äÒâ6ÜCæeåçï ¢ What purpose does it serve to measure formulas rather than the under- (S¥ â â! K j !( äy¤ !⯠K¡ ¡ ª ( lying set? As algebraic geometers are eager to remark, each formula defines æeÛCÜCÙÒÜbâÄáCâ6æeäÒâ6á êå;ÚeàÜ æeÛ5£ àÒéqî îMâúè6æe× ábæeå;ÛCâúÛCéeêâ6ÜCàÒåç×ÒÚÜCéÜCàÒâq:{ÜCà ë©éqîâ6á ø¥¤uÙÒåç褣uï;ó§û Ø¦× an infinite collection of sets. For instance, for each finite field Fq, we can ÿ ÿ ÿ ÿ Sî<àÒâ6áCâ ó ø¥¤§ÙÒå;褣uïçóuû îMâêâæe×ÜCàÒæeܦÜCàÒâ ×ÙÒê â6áéeþÛCÜbâ6ëÒÛåçÛ éeÙÒ×ÒäÒâ6ä óÃæ ë¨éîMâ6áéeþAïçéeÚ : ' # take the set of F points on the circle: ¢ q ÿ ÿ ÿ àÒå;Û<ëÒáCé ï;â6ê îMæeÛ â6æeÙÒÜCåçþ.ÙÒï;ïçó§Ûbéeï¥ eâ6ä ó è6éeêëÒÙÒÜCâ6á>Ûbè6å;â6×ÜCåçÛCÜCÛXïçée×ÒÚæeÚeéwJ ¨ 2 2 2 #¥u v w&'yx#Sêé§ä}#¥:!~yuo W'5'E¤§ÙÒå;褣uïçó âM¦Ò×Òâ ó¤t#¡ub'# #¥u v wP'jæe×Òä âî¢åçÛCàÜbéãè6éeêëÒÙÒÜCâ (4) {(x, y) ∈ F | x + y = 1}. ¢© Ì q ÜCàÒâ6×$ó;@{ô#¥u{'Ëz óG&#¡ub'\[º#Sêé§ä}#¥:!~yuGW'5'>þ.éeá<k< Þç#SæeÜ>âæeèQàÛbÜCâ6ëîMâäÒâÜCâ6áCêå;×ÒâdóG #¥u{'\[¦æe×Òä Uu{G jÛCéÜbàÒâäÒâ6Úeábâ6â éeþ{ÜbàÒâáCâÛCÙÒï;ÜCåç×ÒÚë©éeï;óu×Òéeêå;æeï&åçÛss@¨ßuæe×ÒäÜCàÒâ6× áCâ6äÒÙÒè6â ÜCàÒâ6×ãáCâ6äÒÙÒè6âêéuä We will see that the motivic measure of the formula is a universal measure ÿ k:ôÜCéé ÜCæeåç׺óG©{ôh' ð¦éeÜCâÜCàÒæeܵó;&#¥u{'zZ#¥uUvfw&'\[ç#Sêé§ä}#¥:!~yu W'5' êé§ä in the sense that the value it attaches to the formula does not commit ¢ ¢ us to any particular field. And yet if we are supplied with a particular field, it will be possible to recover the traditional measure of a set from the motivic measure of its defining formula. In this sense, motivic measure is to traditional measures what an algebraic variety is to its set of solutions. 2. Counting measures and Finite Fields Counting is the fountainhead of all measure. The measure of a finite set is its cardinality. At the risk of belaboring the point, in preparation for what 4 THOMAS C. HALES m .kü.Õ ¦MÕ©¨ k k k ¦ Ó k Ó .Õ £ £ £ is to come, let us recast ordinary counting. The scissor relation for disjoint ÿ ÿ ï;â ÜCé þ.æeè6ÜCéeá×ÙÒê â6áCÛå;× ë¨éeïçóu×Òéeêåçæeï{ÜCåçêâ/ßæe×Òä ÜCàÒå;Ûå;Ûj×Òéî ÜCàÒâ finite sets is ×Òé§å;äÒâ6æ î¢àÒâ6ÜbàÒâ6áåçÜjå;Ûjë¨éeÛCÛbå ÿ éeÙÒÜCÛCÜbæe×ÒäÒå;×ÒÚÄëÒáCé ïçâ6ê éeþ{ÜCàÒåçÛ¢æeáCâæ ¢ ÿ éeÙÒÜëÒáCå;êæeï;å;ÜFó ÜCâ6ÛbÜCå;×ÒÚ&å ÖFþ¦ÛCéeêâée×ÒâÚeå¥ eâ6ÛóeéeÙÆæe×xå;×ÜCâ6Úeâ6áæe×ÒäxæeÛCÛCâ6áCÜbÛÄÜCàÒæeÜÄåçÜ [X ∪ Y ] = [X] + [Y ]. àÒæeÜÄæ Ì § ÿ å;ÛëÒáCå;êâ/ß$èæe× óeéeÙíèàÒâ6褣ÜCàÒæeÜÜCàÒå;Ûå;ÛÛCéå;Ûå;×íë©éeï;óu×Òéeêå;æeï2ÜCåçêâMå æe×ÜCàÒâ6ó Úeå¥ eâóeéeÙ â6ÜCÜbâ6á § ÿ M §å;äÒâ6×Òè6âÃÜCàÒæe×ÜCàÒâå;áÄÛCæóuýSÛCé ÜCàÒæeÜúå;Üå;Ûúæ ëÒáCåçêâ×ÙÒê â6á5å æe×xÜCàÒâ6óÆëÒábé §å;äÒâÛCéeêâÛCéeáCÜúéeþ More generally, if we allow the sets to intersect, it is â ÿ øè6â6ábÜC奦Òè6æeÜbâ6ûüÜCàÒæeܧÚeå¥ eâÛ§óeéeÙ¡æeï;ï¦ÜCàÒâ å;×Òþ.éeáCêæeÜCå;ée× óeéeÙ ×Òââ6ä ÜCé eâ6áCå;þ.óÜCàÒæeÜ ÜCàÒâ ×ÙÒê â6áåçÛ ÿ ëÒáCå;êâMå ÖÜå;ÛÄ×ÒéeÜwßæeÛÄþ.æeáÄæeÛÄÖ è6æe×ÛCââ/ßMé §å;éeÙÒÛÄàÒéî ÜbéíäÒé ÛCéÄ<èâ6áCÜCæeåç×Òï;óô×ÒéeÜÄæeÛ (5) [X ∪ Y ] = [X] + [Y ] − [X ∩ Y ]. å;×ÒäÒââ6äæ ÿ ÿ é uå;éeÙÒÛ¢æeÛXî<å;ÜCà ÜCàÒâþ.æeè6ÜCéeábå;×ÒÚÄëÒáCé ïçâ6ê ÖFÜ<ÜCÙÒáC×ÒÛXéeÙÒÜ¢ÜCàÒæeÜ>ÛCéeêâéeï;äãáCâ6êæeáy£uÛ>éeþÝ$ÙÒè6æeÛXþ.ábéeê ¢ ÿ k¤&&ëeÞw:ZÛGè6æe× âêéuäÒ奦Òâäþ.éeá>ÜCàÒåçÛ¢ëÒÙÒábë¨éeÛCâKJ The congruence relation asserts that ÜCàÒâ 2åçáCÛCÜ{×ÒéeÜCâÜCàÒæeÜH:åçÛëÒáCåçêâMåçþÒÜCàÒâ6áCâBæeáCâëÒáCâ6è6åçÛCâ6ï;ó/: åçקÜbâ6Úeâ6áCÛ!w¦å;×ÜbàÒâMáCæe×ÒÚeân/rw¨r : 0 n: àÒâ6áCâþ.éeáCâ¦å;þ&îMâ¦èæe×úÛCàÒéqî¡ÜCàÒâXâ6ö§å;ÛCÜbâ6×Òè6â¦éeþ:d >ÛCÙÒèà§å;×ÜCâ6Úeâ6ábÛMÜCàÒâ6× [X] = [X ]. î<àÒå;èà æeáCâXè6éeëÒáCåçêâXÜbé ¢¨ îMâ¢àÒæ eâ¢æëÒáCéuéeþ¨ÜCàÒæeÜ: å;ÛëÒábå;êâ Ö×þ.æeè6Üåçþt:å;Û¥ëÒáCåçêâXÜCàÒâ×ÄÜCàÒâ6ÛCâ/ qæeïçÙÒâ6Ûþ.éeáCê æ èóuè6ïçå;è>ÚeábéeÙÒë$ß ¢ [ xaF{[ eâjæ Úeâ6×Òâ6áCæeÜCéeáImħÜCàÒæeÜBåçÛ]ߧÜCàÒâáCâjâ6öuåçÛCÜCÛGæe× å;×ÜCâ6Úeâ6áq þ.éeáBî<àÒå;èà=^~QH~Q ~-õ-õ-õ\~Q æeáCâ 0 æe×Òä§ÛCéàÒæ whenever there is a bijection between X and X . The scissor group Scount ÿ æeï;ïªè6éeëÒáCåçêâ¦ÜCén:íæe×Òä äÒåçÛCÜCå;×ÒèÜMêé§äq: àÙÒÛBÜCé ÛCàÒéî ÜCàÒæeÜ:íåçÛMëÒáCåçêâXîMâ¦×Òââ6ä Ûbå;êëÒïçóâöuàÒå åçÜ is the quotient of the free abelian group on finite sets satisfying the scissor ¢¨ ÿ ãæe×ÒäëÒábé eâÜCàÒæeÜ¢ÜbàÒâ6ÛCâ×ÙÒê â6ábÛXæeáCâäÒå;ÛCÜCåç×Òè6Ü>êéuäU: ÖF×ãþ.æeè6ÜWå;Û<æÚeâ6×ÒâáCæeÜCéeá>åçþæe×Òäée×Òïçó å;þ and congruence relations. It is is isomorphic to Z. The cardinality #X of a ¢ ÿ ÃàÒæeÛ¦éeáCäÒâ6á:º=#Sêéuäk: ' Öܦèæe× â ÛCàÒéî¢× ÜCàÒæeܦæe×óÛCÙÒèà éeábäÒâ6ájê ÙÒÛCܦäÒå¥ §å;äÒân:ºeß©æe×Òä ¢ ôc xaF ÛCé ée×Òâ è6æe×ÃÛCàÒéî ÜCàÒæeÜXåçþ åçÛ¦×ÒéeÜXæ Úeâ6×Òâ6áCæeÜCéeá¦ÜbàÒâ6×' ¸@ SzüC#Sêé§ä7'¦þ.éeá¦ÛCéeêâ ëÒáCå;êâ finite set X factors through the scissor group b äÒå¥ §å;äÒåç×ÒÚo:µº àÙÒÛAæøè6â6ábÜC奦Òè6æeÜbâ6ûÜbéÛbàÒéqîxÜCàÒæeÜ{:åçÛ¥ëÒábå;êâGîMéeÙÒïçäèée×ÒÛCå;ÛCÜ éeþæe×Òän¦¢ëÒáCå;êâ ¢¨ ô xaF äÒå¥ §å;äÒâ6Û:É+p}ßÒæe×ÒäãÜCàÒâèàÒâ6褣eâ6áXîMéeÙÒïçä×Òâ6â6äÜbék eâ6áCåçþ.ó§ÜbàÒæeÜ< z =#Sêé§äk: '<î¢àÒâáCâ6æeÛ X 7→ [X] 7→ #[X] ∈ Z. J< ôc x F ÿ ¸© zZ=#Sêé§ä7'Gþ.éeáGæeï;ïëÒábå;êâÛäÒå¡ uå;äÒåç×ÒÚY: feߧÛbéeêâ6ÜCàÒåç×ÒÚÜCàÒæeÜBèæe× âjæeè6èéeêëÒï;åçÛCàÒâ6ä b Of course, if our only purpose were to count elements in finite sets, this å;× ë¨éeïçóu×Òéeêåçæeï&ÜCå;êâ ¢ ÿ ï;âê ÜCàÒéeÙÒÚeà Jmæ×Òâ<×Òââ6äÒÛAèâ6áCÜC塦Òè6æeÜCå;ée× ÜCàÒæeÜ2â6æeèà ÛCÙÒèàD¢åçÛ¥ëÒábå;êâ àÒâGÛCéeïçÙÒý construction is overkill. The first motivic measure that we present is an àÒâ6áCâ<å;Û2æëÒáCé ¨ ¢¨ ÿ é§ eâ¦æeï;ÚeéeáCåçÜCàÒêÄæe×Òä ée×Òâjè6æeקÛCàÒéî ÜCàÒæeÜB×Òé êéeáCâ¦ÜCàÒæe×C#SïçéeÚ: 'h»#Tï;éeÚ Ý&' analogue of this approach to counting. We call it the motivic counting ÜCåçéeקåçÛGÜCéå;ÜCâ6ábæeÜCâ¦ÜCàÒâjæ ÿ éuäÒäÃëÒáCåçêâ6Ûj×Òâ6âä ÜCé â è6â6áCÜb奦Òâ6ä æeþ.ÜCâ6á¦ée×ÒâàÒæeÛ¦å;Übâ6áCæeÜCâ6äÃæeïçïÜCàÒâ îMæóäÒéî¢× àÙÒÛîâ àÒæ§ eâ æ measure. The scissor relation will be similar to Equation 5. ¢/¨ ë¨éeïçóu×Òéeêå;æeï(ÜCå;êâXè6â6ábÜC奦Òè6æeÜbâq#SÛCàÒéeáCÜëÒáCéuéeþR'ÜCàÒæeÜ:íå;ÛMëÒáCåçêâ/ß/æe×ÒäúÛCé ëÒáCåçêæeï;åçÜFóÜCâÛCÜCå;×ÒÚ å;ÛMåç×úÜCàÒâ è6ïçæeÛCÛ({¡Ê ¢ ÍGÙÒÜ åçÛC× :ZÜÜCàÒåçÛ ÜCàÒâæeï;ÚeéeáCåçÜCàÒê îMâÛbâ6âM£Kå éuâ6ÛC× : Ü ÜCàÒåçÛÚeå¡ eâæãë¨éeï;ó§×Òéeêåçæeï2ÜCå;êâæeïçÚeéeáCå;ÜCàÒê 2.1. Ring formulas. Traditional measure calls for a full discussion of the ÿ þ.éeáBäÒâ6ÜCâáCêåç×Òå;×ÒÚî¢àÒâÜCàÒâ6áBæ Úeå¥ eâ×úå;×ÜCâ6Úeâ6á;:íåçÛBëÒáCå;êâMå àÒâ¦æe×ÒÛbîMâ6áGå;Ûø×Òéeû â6è6æeÙÒÛCâjæeïçée×ÒÚ ÜCàÒâ class of measurable sets. Since we work with formulas rather than sets, our ¨ îMæó îMâ îMéeÙÒïçäãàÒæ§ eâ ÜCéþ.æeè6ÜCéeá¯:Én¤uÙÒåçèh£§ï;óߧÛbéeêâ6ÜCàÒåç×ÒÚÄ×Òéeýée×ÒâE£u×Òéî¢Û<àÒéî ÜbéäÒé approach calls for a full discussion of the class of formulas to be measured. ¢ (5( ã â!j! ã} ª¡ ã} º § £¡êã}ä-( We allow all syntactically correct formulas built from a countable collec- ÖF×ÃÛCâ6è6Übå;ée× %Äîâ å;×ÜCáCéuäÒÙÒèâ6ä ÜbàÒâ×ÒéeÜCå;ée×éeþøåç×ÒäÒÙÒÛCÜCáCåçæeïÛCÜCábâ6×ÒÚeÜCàÃëÒáCå;êâ6ÛCû Ö×Ãþ.æeè6Ü>å;þ éeÙÒá ¢ tion of variables xi, parentheses, and the symbols ¢ ÿ ÿ Úeå¥ eâ×ãå;×ÜCâ6Úeâá¢åçÛ¢è6éeêë¨éeÛCåçÜCâÜCàÒâ6×ãÜbàÒâ6áCâ åçÛ¢æ ëÒáCé æ åçï;å;ÜFó éeþæeÜGïçâ6æeÛCÜ©^»qÝÜCàÒæeÜ<â6æeèàãæeëÒëÒïçå;è6æeÜbå;ée× ÿ ëÒáCé§ uåçäÒå;×ÒÚ æ ÛCàÒéeáCÜjè6â6ábÜC奦Òè6æeÜbâ¼ eâáCå;þ.ó§å;×ÒÚ ÜCàÒæeÜjÜCàÒâ ×ÙÒê â6á (6) ∀, ∃, ∨, ∧, ¬, 0, 1, (+), (−), (∗), (=) éeþ¥ÜCàÒæeÜãøî¢åçÜC×Òâ6ÛCÛbû ÜCâÛCܦÛCÙÒè6è6ââ6äÒÛ åç× å;Û<è6éeêë¨éeÛCåçÜCâd#SÜCàÒâè6â6áCÜC塦Òè6æeÜCâëÒáCé§ uå;äÒâÛ¢æî¢åçÜC×Òâ6ÛCÛ`wP' àÒå;Û<å;Û<漩y¬&´b®³&dz&¥Æ"´t³&ÇU«¥¬&!§«¥Ç¿BÜCâÛCÜ ¢ ¨ þ.éeáXè6éeêë¨éeÛCåçÜCâ6×Òâ6ÛbÛU#Tè6éeêëÒïçâ6öuåçÜó§è6ïçæeÛCÛx¡Ê' ئÛXîMâ ×ÒéeÜCâäå;ÜXåçÛXæeï;êéeÛCÜ>è6âáCÜCæeå;×ÃÜCéëÒáCé §å;äÒâÛbÙÒèQà More precisely, we allow all formulas in the first-order language of rings. A ¢ æ ëÒáCéuéeþ åç× ÞeÞúáCÙÒ×ÒÛjéeþ¥ÜCàÒâÜCâ6ÛCܦåçþb:åçÛ¦å;×ÒäÒâ6âäè6éeêë¨éeÛCåçÜCâ/ß©ÛCé§å;þ åçܦþ.æeå;ï;Û¦ÜbàÒâ6× åçÜXå;Ûµ eâáCóï;å¡£eâ6ï;ó ÿ :íåçÛMëÒáCåçêâ æÙÒáBêæeå;×úé ñâ6èÜCå;ée×úîMæeÛBÜbàÒæeÜÜCàÒåçÛMäÒéuâ6Ûb× :ZÜBëÒáCé§ uåçäÒâXæ ëÒáCéuéeþ&ÜbàÒæeÜ:íå;ÛMëÒábå;êâ formula that has been constructed from these symbols will be called a ring ÜCàÒæeÜ ¢ ¢ ÿ ×Òâúé ñâ6è6ÜCå¡ eâ/ß(ñÙÒÛCÜÛbàÒéeáCÜéeþg¦Ò×ÒäÒå;×ÒÚæãë¨éeï;ó§×Òéeêåçæeï¥Übå;êâÜCâ6ÛCÜþ.éeáëÒáCå;êæeï;åçÜóß&å;Û ÜCéɦÒ×Òä æ formula. We avail ourselves of the usual mathematical abbreviations and æ ÿ ë©éeï;óu×Òéeêå;æeïMÜbå;êâ ÜCâÛCÜþ.éeá ëÒábå;êæeïçå;ÜFó àÒåçÛ îæeÛ æeèàÒåçâM eâ6ä ó ئäÒï;âêæe×ôæe×Òäsþ¦ÙÒæe×ÒÚ n áCæe×ÒäÒéeê renamings of variables. We write 3 for 1 + (1 + 1), x for x ∗ x ∗ x · · · ∗ x (n ¢¨ sWñ&ñeÝ ÙÒÛCå;×ÒÚ æ êâ6ÜCàÒé§ä éeþ¥è6éeÙÒ×ÜCå;×ÒÚ ë¨éeå;×ÜCÛjée× â6ï;ïçå;ëÒÜCåçè æe×Òä àóuë¨âáCâ6ï;ïçå;ëÒÜCåçèè6ÙÒá5 eâ6Ûé eâáE¦Ò×ÒåçÜCâ times), xy for x ∗ y, a + b + c for a + (b + c), and so forth. å;× ÿ ÿ ¦Òâ6ïçäÒÛ # æeÛCâ6ä ée× å;äÒâæeÛéeþ!k éeïçäÒîMæeÛCÛCâ6á æe×Òä òå;ïçå;æe×t' ئï;ÜbàÒéeÙÒÚeà â6æeÙÒÜbå;þ.ÙÒïMåç× ÛCÜCáCÙÒè6ÜbÙÒáCâ/ßÜCàÒâ6åçá ¢ ÿ o eâ6áCó§è6éeêëÒïçå;è6æeÜbâ6äæe×ÒäæeïçêéeÛCÜ¢è6â6áCÜCæeåç×Òï;ó å;êëÒáCæeè6ÜCåçè6æeï4ß}æeÛXîâ6ï;ï"æeÛ â6å;×ÒÚúábæeÜCàÒâ6á>äÒåè6ÙÒïçÜ With usual abbreviations, ÜCâ6ÛbÜ>å;Û ÜCéMñÙÒÛCÜCåçþ.óÜCàÒâ6éeábâ6ÜCå;èæeï;ï;óå;×æeï;ïuå;ÜCÛ{äÒâÜCæeå;ïçÛ ÖFÜ2äÒéuâ6Û2àÒéîMâM eâ6á ëÒáCé§ uåçäÒâGæÛbàÒéeáCÜ2è6â6áCÜb奦Òè6æeÜCâg eâáCå;þ.ó§å;×ÒÚ ‘∀x y z. x3 + y3 = z3’ ¢ is a ring formula, because its syntax is correct. But ‘))∀ + ∀ = 2∀((’ and ‘ ∧ ∨ ∧ ∨ ∧ ’ are not ring formulas. WHAT IS MOTIVIC MEASURE? 5 P ¦ Ó ¦ " ` £m £ 2.2. The scissor group of ring formulas. We imitate the construction of ÿ è6éeêë¨éeÛCåçÜCâïç奣eâÜCàÒåçÛ ÙÒÜ]ß&îMæeåçÜæ§êéeêâ6קÜwß$ÛCåç×Òè6âd%&ë&% å;ÛjâM eâ6×ÛCéãîMâ è6æe×íÜbæ&£eâ ÜbàÒâÛ5¤§ÙÒæeáCâ6áCé§éeÜ the scissor groups Spoly and Scount to build the scissor group of ring formulas. [ Ñ æeÚeæeå;×$ßuæe×Òäãæè6æeïçè6ÙÒï;æeÜCåçée×ãáCâW eâ6æeï;Û¢ÜCàÒæeÜ£&% z¡ÝeÝeÞ|#Têéuä6ñ&%&ñ&']ßuÛCéÄÜbàÒæeÜñ&%&ñå;Û<è6éeêë©éeÛCå;Übâ ÖF× ¢ Take the free abelian group on the set of ring formulas. ÿ Úeâ6×Òâ6ábæeï4ß}å;×ÜCâ6Úeâ6áµwå;Û<æ8*«±§ª´b¿R²y²XÜCéE: â6å;×ÒÚÄèéeêë¨éeÛbå;ÜCâå;þÜCàÒ⩦Ò×ÒåçÜCâ ÛCâW¤uÙÒâ6×Òè6â We impose two families of relations. The scissor relation takes the form ô ôc ôc xaF xaF [ xaF [d ¸ ¸ #Sêé§äk: 'h~Ywab #Sêé§äk: 'h~-õ-õ-õh~Ywab #Sêé§äk: ' w established in Equation 5 for unions. Scissor relations. If φ ∨ φ is a disjunction of two formulas, then Sî<àÒâ6áCâo: |(ùÝBegfãî¢åçÜCà¢fãéuäÒät'GåçÛ<×ÒéeÜGâM¤§ÙÒæeï&ÜCéâ6å;ÜbàÒâ6áE^~ ^~-õ-õ-õh~ ¦éeáY^~ ^~-õ-õ-õ¤~ ^~-¼^~;h&~-õ-õ-õ8~;h # 1 2 ¢ ÖFÜBå;Ûg£§×Òéqî<קÜCàÒæeÜ]ߧþ.éeáBæeïçï¨éuäÒä§è6éeêë¨éeÛCåçÜCâË:GߧæeÜMïçâ6æeÛCÜBÜCàÒáCââ6ýJ¤§ÙÒæeáCÜCâ6áCÛGéeþ$ÜCàÒâ¦å;×ÜCâ6ÚeâáCÛ/wK~S`r (7) [φ1 ∨ φ2] = [φ1] + [φ2] − [φ1 ∧ φ2]. w¨rs:üæeáCâî<å;ÜC×Òâ6ÛbÛCâ6Û>þ.éeá: õÒéÄè6æe×ãî⩦Ò×Òäæî¢åçÜC×Òâ6ÛCÛ ø¥¤uÙÒå;褣§ï;óuû å;þA:÷åçÛ¢è6éeêë¨éeÛCåçÜCâMå ¢ ÿ æ×ÒâGå;äÒâ6æ¦åçÛ{ÜCé¦ÜCáCóSwiùÝ"~j¥"~\%"~-õ-õ-õè6ée×ÒÛCâ6èÙÒÜCå¥ eâ6ïçóÙÒ×ÜCå;ï§îMâG¦Ò×Òä æ¦î¢åçÜC×Òâ6ÛCÛ â â6ïçå;âM eâÜCàÒæeÜ l ¢Ì Er Ý#Sï;éeÚm: '\[qߪÜCàÒéeÙÒÚeàÃîMâ è6æe×Ò×ÒéeܦëÒáCé§ eâ ÜCàÒå;Û>â6öuè6âëÒÜÙÒ×ÒäÒâá¦ÜCàÒâ æeÛCÛCÙÒêëÒÜCå;ée× ÜCàÒâ6ábâ å;Û¦æúî<å;ÜC×Òâ6ÛbÛ To describe the congruence relation, we must decide what it should mean ÿ éeþæ å;ÚÄÜbéuéeï4ßÜCàÒâikâ6×Òâ6áCæeïçå¥ 6â6äml>å;âêæe×Ò×Éþ¦ó§ë¨éeÜCàÒâ6Ûbå;Û ¢ for two ring formulas to be congruent. By way of analogy, in the case of Þå;褣 åçקÜCâÚeâ6áCÛ w åç×on^~¢Ý"~j¥"~-õ-õ-õ8~y:qpæeÜáCæe×ÒäÒéeê ÙÒקÜCåçïGîMâ6¦Ò×Òäüæî<å;ÜC×Òâ6ÛbÛ Í<ó î<àÒæeÜ îMâ l ¢ ÿ ÿ ÿ é§ eâ/ߧåçþ: åçÛBè6éeêë©éeÛCå;Übâ¦ÜCàÒâ6קÜCàÒâ¦ëÒáCé æ å;ïçå;ÜFóÜbàÒæeÜB×Òée×Òâéeþ"ÜCàÒⵦÒábÛCÜ rÄåçקÜbâ6Úeâ6áCÛBèàÒéeÛCâ6× î<áCéeÜCâ¦æ polygons, two are congruent if there is a bijection between the two sets that ÿ ÿ is ^»%Bt àÙÒÛ î<å;ÜCà æ§à§ÙÒ×ÒäÒáCâäíéeáÛbé ÛCÙÒèàíÜbâ6ÛCÜCÛîMâÚeâ6Üæ ëÒáCé æ åçï;åçÜóãÜCàÒæeÜ æeáCâî<å;ÜC×Òâ6ÛbÛCâ6Û åçÛ is induced by an isometry. Our first guess at the congruence relation for ¢Y¨ ÿ ÿ ¡ qæ ïçâÜCàÒæeܦå;ܦè6éeÙÒïçä é§è6è6ÙÒájå;×ÃëÒáCæeèÜCå;è6â&Ä&Ûbé îMâ â6ï;åçâM eâÜCàÒæeܦæe×ó å;Û¦Ûbé ÛCêæeï;ï{ÜbàÒæeܦå;ܦå;Û¦åç×Òè6ée×Òè6â6å ring formulas is that two ring formulas are congruent if there is a bijection å;×ÜCâ6Úeâá:ôþ.éeá¢î<àÒå;èà×Òée×Òâéeþæà§ÙÒ×ÒäÒáCâäáCæe×ÒäÒéeêïçó§èàÒéeÛCâ×çw3: Û¢åçÛ<æî¢åçÜC×Òâ6ÛCÛwßÒåçÛ¢ëÒábå;êâ âè6æeï;ï ¢;Ì between the sets of solutions for each finite field Fq. (We limit ourselves to ÛCÙÒèà : øÖ«¡´b®tÛ²^§ª©5«¡¬&{²^§ª©^¿R´tÚ§Ømt©5«¡Ç¿R²nû ¢ ÿ ÿ ÿ ÿ å;ÚúëÒáCé ïçâ6ê î<å;ÜCàÃÜCàÒâ æ é eâ êâ6ÜCàÒéuäÃåçÛXÜCàÒæeܦæeï;ÜbàÒéeÙÒÚeàîMâÛCÜCáCée×ÒÚeïçó âï;å;âW eâÜCàÒæeܦæe× àÒâ finite fields because we are attempting to imitate the counting measure of ¨ eâú×ÒéÃëÒáCé§éeþßæe×ÒäôêæeÜCàÒâ6êæeÜCå;è6åçæe×ÒÛ ï;å¡£eâ å;×ÒäÒÙÒÛbÜCáCå;æeïÛCÜCáCâ6×ÒÚeÜbàôëÒábå;êâúåçÛå;×ÒäÒââ6ä÷æëÒábå;êâeß2îâúàÒæ finite sets.) However, there are two modifications that we must make to this ëÒáCé§éeþ ÖF×ÒäÒâ6â6äåçþ{óeéeÙè6ïçæeå;ê ÛCÙÒèàå;×ÜCâ6Úeâ6ábÛ>æeáCâëÒáCå;êâ/ß}î¢åçÜCàÒéeÙÒÜ>ëÒáCé§éeþß}ÜCàÒâ6×æèóu×Òå;èêåçÚeà§Ü<×ÒéeÜ ¢ first guess to arrive at a workable relation. ÿ âï;å;âW eâÜCàÒæeÜóeéeÙÒá áCæe×ÒäÒéeêï;ó èQàÒéeÛbâ6× wæeábâÛCéábæe×ÒäÒéeêßéeáÜCàÒæeÜ óeéeÙíæeábâÙÒ×Òï;ÙÒ褣§óß$éeá ð¦éªß ¢¡¢¥¢ The first modification is to use pseudo-finite fields rather than finite fields. ÿ î<àÒæeÜ¢îMâ×Òâ6â6äåçÛ¢æëÒáCéuéeþÜCàÒæeÜ¢æקÙÒê â6áXåçÛ¢ëÒáCåçêâî¢àÒâ×îMâ ÜbàÒå;×t£ ÜCàÒæeÜ<å;ÜGå;Û ¢ A pseudo-finite field is an infinite perfect field such that every absolutely (5u(E wv±äe =¡ê Pã}â § ! jhPªä-ä©( irreducible variety over the field has a rational point and such that there is § Ø<ÜÜCàÒâ=WñeÞ¦¥ êââ6ÜCå;×ÒÚ éeþ>ÜbàÒâ ئêâ6áCå;èæe×÷ìÃæeÜCàÒâ6êæeÜCå;èæeïBõÒéuè6åçâ6ÜFóßI ð éeïçâ è6æeêâ§ÜCé ÜCàÒâ ¢ ¢ a unique field extension of each finite degree (inside a fixed algebraic closure ÿ ÿ ï;æe褣 éeæeáCä æe×Òäî¢åçÜCàÒéeÙÒÜ<æîMéeáCä î¢ábéeÜCâäÒéî<× of the field). The defining properties of a pseudo-finite field are properties ß8Ñ ! Wñ¦¥&ëeÞ&ë&ëeÝx"ºë¦£¤&¦¥¦&eݦ.&ëeݦ&&ë"~ Ý possessed by finite fields (except the part about being infinite). Moreover, logicians have found that the behavior of pseudo-finite fields is essentially no ÿ â6áCÛéeÙÒÜjée× ÜCàÒâáCåçÚeà§Ü¦Ûbå;äÒâéeþAÜbàÒâ âW¤uÙÒæeÜCåçée×$ߨæe×Òä äÒâ6Übâ6áCêåç×Òå;×ÒÚ ÜCàÒâ ï;ée×ÒÚeýSê ÙÒï;ÜCåçëÒï;ó§å;×ÒÚ ÜCàÒâקÙÒê different from the generic behavior of finite fields, but they avoid the hassles äÒâ6è6åçêæeïâöuëÒæe×ÒÛCåçée× éeþ$ÝKß8Ñm XÜCéëÒáCé§ eâÜbàÒæeÜBàÒâîæeÛBå;×ÒäÒâ6â6ä è6éeáCáCâè6Ü Ø¦þ.ÜCâ6áCîMæeábäÒÛ<àÒâjÛCæeåçä ÜCàÒæeÜ ¢ ÒÚeÙÒáCåç×ÒÚãÜCàÒåçÛéeÙÒÜjàÒæeä ÜCæ&£eâ6× àÒå;ê øÜCàÒáCââóeâ6æeábÛéeþAõÒÙÒ×ÒäÒæqó§ÛCû àÒâ êéeáCæeï2éeþ¥ÜCàÒå;ÛjÜCæeïçâ åçÛjÜCàÒæeÜ ¦ that appear in positive characteristic. For those seeing pseudo-finite fields ¢©¨ § &£ éeïçâ æ ÚeáCâ6æeÜäÒâ6æeï¥éeþîMéeá5£Ãæe×Òä ë¨â6áCÛbâM eâ6áCæe×Òè6âÜCé*¦Ò×ÒäíÜbàÒâ6ÛCâÄþ.æeè6ÜbéeáCÛ]ß&å;ÜäÒå;ä æeï;ÜbàÒéeÙÒÚeàå;ÜÜCé§é for the first time, it would not be a severe distortion of the facts to ignore &£eâàÒå;ê ïçée×ÒÚ ÜCé ñÙÒÛCÜCåçþ.óÃàÒå;ÛáCâ6ÛCÙÒïçÜÜbé æ§áCéuéeê þ.ÙÒïçï2éeþêæeÜCàÒâ6êæeÜbå;è6åçæe×ÒÛU#Tæe×Òä$ß&åç×ÒäÒâ6â6ä$ß&ÜCé ×ÒéeÜjÜCæ the ‘pseudo’ and to work instead with finite fields. Úeå¥ eâjæëÒáCéuéeþÜCàÒæeÜ¢àÒâîMæeÛ<è6éeáCáCâè6Ü5' àÙÒÛ>îâ ÛCâ6âÜCàÒæeÜ<ée×Òâèæe×ëÒáCé§ uå;äÒâæ ÛCàÒéeáCÜ<ëÒáCéuéeþß}âM eâ6×ãå;þ ¢¨ The second modification is to require that the bijection between the solu- ¦Ò×ÒäÒå;×ÒÚÜCàÒæeÜ¢ëÒábéuéeþÜbæ&£eâ6Û>æï;ée×ÒÚ ÜCå;êâ ¢ tions come from a ring formula that is independent of the underlying field. ÿ ÖF×Úeâ×Òâ6áCæeï ée×Òâèæe×â6ö§àÒå å;Üjþ.æeè6ÜbéeáCÛéeþAæ Úeå¡ eâ6× å;×ÜCâ6Úeâ6áo:¡ÜCé§Úeå¥ eâæ§ÛCàÒéeáCÜëÒáCéuéeþAÜCàÒæeÜ: åçÛ ç#SÛCÙÒèàüëÒábéuéeþ.ÛæeáCâúè6æeïçï;â6äTÀM¿R©^§«zybÀR¬§5¿¥²8' Í<ó øÛCàÒéeáCÜCû îMâúêâ6æe×÷ÜbàÒæeÜ ÜCàÒâ ëÒáCéuéeþ<è6æe× è6éeêë¨éeÛCåçÜCâ We are now ready to state the congruence relations. ¢ ÿ ÿ ç eâ6áC塦Òâ6ä åçסë©éeï;ó§×Òéeêå;æeï>ÜCå;êâ/ß<æe×Òä îMâ ÛCæqóxÜbàÒæeܧÛbÙÒèQà ëÒáCé ï;â6êÛ æeábâ åçסèï;æeÛCÛ|{"Êp#wøÖ´t³&´t× â Congruence relations. ÿ ®t¿M§5¿¥©5ÇU«¡´t«¥²^§ª«9À#A³&¥Æ"´t³&ÇU«¡¬&˧ª«¥ÇÉ¿[û&' â æeáCâ×ÒéeÜÛbÙÒÚeÚeâ6ÛCÜCåç×ÒÚ ÜCàÒæeÜjÜCàÒâëÒáCé§éeþAèæe× âþ.éeÙÒ×Òä å;× ¢µÌ 0 ÿ â èàÒâ6褣eâ6ä å;×Ãë¨éeïçóu×ÒéeêåçæeïÜbå;êâ&Ä©å;×ÒäÒâ6âä îMâàÒæ§ eâ ë¨éeïçóu×Òéeêå;æeï$ÜCå;êâ/ß©ée×Òï;ó ÜCàÒæeÜXÜCàÒâ ëÒábéuéeþ è6æe× [φ] = [φ ] } if there exists a ring formula ψ such that for every pseudo-finite field K rQm¥nlFI~@¢| ~IaBXiFnrV¤çogrgt rVvG|¢t]n|v{mIÅg/brVvGv{rg~Srg npIm¥rQyÒbrVny n~n| rg £ of characteristic zero, the interpretation of ψ gives a bijection between the tuples in K satisfying φ and the tuples in K satisfying φ0. Example 2.1. The congruence relation gives [‘∃x. x2 + bx + c = 0’] = [‘∃X.X2 = B2 − 4C’] The formula ψ realizing the congruence and the bijection at the level of points is ‘(b = B) ∧ (c = C)’. 6 THOMAS C. HALES m .kü.Õ ¦MÕ©¨ k k k ¦ Ó k Ó .Õ Ç £ £ £ That is, in every pseudo-finite field of characteristic zero, a monic quadratic ô ªéeá¢â6ö§æeêëÒïçâ ß}ÜCæ&£uåç×ÒÚ wùÝîMâ è6æeïçè6ÙÒï;æeÜbâÜCàÒæeÜ polynomial has a root if and only if its discriminant is a square. ô§ô Ý zZݦ¥ #Têéuä=ÞeÞW'h~ Definition 2.2. The scissor group Sring of ring formulas is defined as the ©£§×ÒéqîùÜCàÒæeÜ©ÞeÞ¦å;Û>èéeêë¨éeÛbå;ÜCâ free abelian group subject to the scissor and congruence relations. ÛCéîMâ ¢ âêåçÚeà§Ü<æeÛ5£ î¢àÒâÜCàÒâ6á>ÜbàÒå;Û<æeï;îMæóuÛ<îMéeá5£§Û ÖF×éeÜbàÒâ6á¢îéeáCäÒÛ]ß ¢ 2.2.1. Counting measure. Ì ÖFÛ>å;ÜGÜCábÙÒâ ÜCàÒæeÜË«¥ÙH: «¥²©ÀR³&ÇA³&²5«9§5¿E§Øb¿R´ : ®³H¿R²/´t³§®«±°^«±®t¿GÝ xS÷Ý&å ªéeá]ß(åçþ2ÛCéªßªîâàÒæ eâ æe eâ6áCó§×Òå;è6â îMæó§ÜCéúäÒå;ÛCÜbå;×ÒÚeÙÒå;ÛbàëÒáCå;êâ6ÛXþ.áCéeê è6éeêë©éeÛCå;ÜCâÛ ×Òþ.éeábÜCÙÒ×ÒæeÜCâ6ïçó Definition 2.3. The counting measure of a ring formula φ is its class [φ] in ¢ ÜCàÒâæe×ÒÛCîMâ6á¢å;Ûúø×ÒéeûÛCåç×Òè6â/ßÒþ.éeá¢â6ö§æeêëÒïçâ/ß the scissor group of ring formulas. ô Ý z¡Ý #Sêéuä ¥&%W'h~ 2.2.2. Fubini and Products. There is a trivial sort of Fubini theorem for ÿ ÿ ÿ ¥&%! &#"$¥ ð¦éeÜbâÜCàÒéeÙÒÚeà ÜCàÒæeÜ ó§ÜCæ&£uåç×ÒÚew%¥æ é§ eâîâ Úeâ6Ü finite sets: the cardinality of a Cartesian product of two sets is the product ÙÒÜ ¢ of the cardinalities of the two sets. To make sense of a Fubini theorem for ô ¥ zZ¤£¦& #Sêéuä'¥&%W'h~ ring formulas, it is necessary to introduce products to the scissor groups; § eâ ÜCàÒæeÜ(¥&%åçÛ¢èéeêë¨éeÛbå;ÜCâ that is, we need a scissor ring. This is easy to arrange. If φ1(x1, . . . , xn) is ÛCéîMâ è6æe×ãÙÒÛCâ ÜCàÒâÛCâ åçäÒâ6æeÛ>ÜCé ëÒáCé ¢ GÙÒÜGÜCàÒâ6× îMâjêå;ÚeàÜGæeÛ5£úî<àÒâ6ÜCàÒâ6á<ÜCàÒå;ÛGæeïçîMæqó§ÛBîMéeá5£§Û]ßuî¢àÒâÜCàÒâ6á<ÜCàÒâ6áCâå;ÛGæeï;îæqó§ÛG²5³&Ç¿Ë qæeïçÙÒâ a formula with free variables x1, . . . , xn and φ2(y1, . . . , ym) is a formula with Í éeþ;wÜCàÒæeÜ<àÒâ6ïçëÒÛ>ÙÒÛ>ëÒábé eâæè6éeêë¨éeÛCåçÜCâ/:ôå;Û<å;×ÒäÒâ6âäè6éeêë©éeÛCå;ÜCâ free variables y1, . . . , ym, and if the free variables of φ1 are distinct from the ¢ free variables of φ , then we declare the product to be ÖF×éeÜCàÒâá¢îMéeábäÒÛ]ß Ë«¥Ù:ï«¥²©ÀR³&Ç)³&²5«±§y¿E§ªØb¿¥´ï§ªØb¿R©^¿ 2 ÖÛ¢å;ÜGÜCáCÙÒâÜCàÒæeÜ +*;Øt«9À±Øe: ®³¿¥²o´t³§o®«±°§«9®t¿`w xS w-, φ (x , . . . , x ) ∧ φ (y , . . . , y ). «¡²o²5³&ÇÉ¿«¡´H§5¿¥Ú¿R©nwdÙ³&© 1 1 n 2 1 m ô ئÚeæeå;×ãÜCàÒâæe×ÒÛCîMâá¦å;Û§ø×ÒéeûúÛbå;×Òè6â+.¦£äÒå¥ uåçäÒâ6ÛYwKÓ8ß |wÄþ.éeáXæeï;ï$å;×ÜCâ6Úeâ6ábÛYw©ß(óeâ6Ü .¦£ /¥"s&0" 1 § éeêë¨éeÛCåçÜCâãåçקÜCâÚeâ6áCÛq:î<àÒå;èàäÒå¥ §å;äÒâçw x6 w þ.éeáÄæeï;ï<å;×ÜCâ6ÚeâáCÛ w æeábâè6æeï;ïçâ6ä21¬&©5ÇU«9À±Øt¬¿¥ This induces a well-defined product on the scissor group Wë ¢ § ÿ ÿ ´AÛÇ ½A¿R©5²]ß3.¦£^~ &Þ¦.æe×Òä WëeÝ&ñ â6åç×ÒÚúÜCàÒâ Ûbêæeï;ïçâ6ÛCÜ¢ÜCàÒáCâ6ââöuæeêëÒïçâ6Û æeáCêåçèàÒæeâ6ï×ÙÒê â6ábÛ¦æeáCâ ¢ y¤uÙÒâ6áCæeäÒåç×ÒÚÃæeÛ ëÒáCå;êâ6Ûï;奣eâÜCàÒå;Û]ß{ÜCàÒéeÙÒÚeà è6éeêëÒÙÒÜCæeÜCåçée×Òæeï;ï;ó ÜCàÒâ6óée×Òï;óíæeëÒë©â6æeá (8) [φ1(x)][φ2(y)] = [φ1(x) ∧ φ2(y)]. æÃקÙÒåçÛCæe×Òè6â/ß2êæeÛ áCæeáCâï;ó ×Òþ.éeáCÜbÙÒ×ÒæeÜCâ6ï;óåçÜjîMæeÛábâ6è6â6×ÜCï;óÃëÒáCé§ eâ6ä ÜCàÒæeÜjÜCàÒâ6áCâ æeáCâ å;×t¦Ò×Òå;Übâ6ï;óêæeקóéeþAÜCàÒâ6êßæe×Òä Under this product, the scissor group becomes a ring. Equation 8 asserts ¢ ÜCàÒæeÜ<î¢àÒâ×îMâ ÚeééeÙÒÜ¢þ.æeá<â6×ÒéeÙÒÚeàÜbàÒâ6ó æeáCâ ×ÒéeÜ<ÛCéábæeáCâ æeÛ<å;Ü˦ÒáCÛCÜ¢æeëÒë¨â6æeábÛ that counting measure satisfies a rather trivial Fubini theorem for ring for- ¢ (54(÷£-6 í &¡ä-( ÖF× æ ¦Òâ6ï;ä$ß&æ§×Òée×ÒýJ 6â6ábéãë¨éeïçóu×Òéeêå;æeïéeþAäÒâ6ÚeáCâ6âk¹§àÒæeÛjæeÜjêéeÛCÜ©¹§ábéuéeÜCÛ mulas – at least for ring formulas without any shared free variables. ¢ [ ªéeáÜCàÒâ ëÒæeáCÜCåçè6ÙÒï;æeáâ6ö§æeêëÒï;âEu ÜbàÒå;Ûå;êëÒïçå;â6ÛÜCàÒæeÜU àÒæeÛBñÙÒÛCÜ ÜFîMé Û5¤uÙÒæeábâ6áCéuéeÜbÛêéuä87"ß&æ ëÒáCåçêâqX¡Ýªßu×Òæeêâ6ï;ó$jæe×Òäç¼ 2.2.3. The universal nature of the counting measure. The counting measure ¢ [ [ [ [ ÖFþMîMâ è6ée×ÒÛCåçäÒâ6á é§äÒäè6éeêë¨éeÛCåçÜCâE:¡ÜCàÒâ6× îM⼤uÙÒå;褣§ï;ó¦Ò×Òä| z % z & z W% #Sêé§ä}¤.&']ß ÿ ÿ 5¤uÙÒæeábâ6áCéuéeÜbÛéeþ`=#Sêéuä}¤.&' Ö× Úeâ6×Òâ6áCæeï2åçþé§äÒäÉ:å;ÛäÒå¥ uåçÛCå ïçâ óÃÜFîMé [φ] of a ring formula φ is designed to be the universal counting measure for ÜCàÒæeÜjå;Ûwß"ÜCàÒâáCâæeábâþ.éeÙÒáÛ ¢ eâ ¬§n9¿R¬&²^§jþ.éeÙÒáäÒåçÛCÜCå;×ÒèÜÛ5¤§ÙÒæeáCâ6áCé§éeÜCÛéeþS=#Sêéuäk: ' à§ÙÒÛîMâ ring formulas. For every finite field F , there is a counting measure on ring äÒå;ÛbÜCå;×Òè6Ü ëÒáCåçêâ6ÛÜCàÒâ6×ÆîMâ àÒæ q ¢ï¨ ÿ eân:ÆåçÛXè6éeêë¨éeÛCåçÜCâ óɦÒ×ÒäÒå;×ÒÚ æúÛ5¤§ÙÒæeáCâ6áCé§éeÜXéeþo=#Sêé§äk: ']ß(î¢àÒåçèà åçÛX×Òâ6åçÜCàÒâ6á formulas: êåçÚeà§Ü>ÜCáCóãÜbéúëÒáCé ×Òéeá`q ¢ n Fq ô ÿ ð¦éîß ó9ªâ6áCêæeÜ;:ZÛ¼ÝåçÜCÜCï;â àÒâ6éeáCâ6êßAå;þ<7÷å;Û ëÒáCåçêâãÜbàÒâ6× #±w>=@?BA '\[D w¦EGF zè}#Sêé§ä)7'wß (9) φ 7→ # (φ) = #{(x , . . . , x ) ∈ F | φ (x , . . . , x )}. C q 1 n q 1 n ¨ ÿ ÛCé w #Sêéuä)7'å;ÛæíÛ5¤uÙÒæeábâ6áCéuéeÜéeþ¼ êéuä)7Ææe×Òäxê ÙÒÛCÜ âï éeá q àÒâáCâ6þ.éeáCâåçþEwIH =@?BA ?BA C C ¢ ¨ Sêéuäk: '¥å;ÛA×Òâ6å;ÜbàÒâ6án<×Òéeág¼<ÜCàÒâ6×q:åçÛAè6éeêë¨éeÛCåçÜCâ Ý$â6ÜG:ZÛAÜCáCóæe×Äâ6ö§æeêëÒïçâKJ â>àÒæ§ eâ#£&% L zZ It gives the number of solutions to the ring formula over a particular finite # NM ¢ Ì #Sêéuä*ñ&%&ñ&'wß(æe×ÒäÃÜCàÒâ Û5¤§ÙÒæeáCâ6áCé§éeÜ £&% Ñ z =#Sêé§ä*ñ&%&ñ&' þ¦êêêêߪîMâ þ.æeåçï;â6äÜCéúëÒábé eâSñ&%&ñúåçÛ field. In contrast, the counting measure of a ring formula takes values in a scissor ring whose construction bundles all pseudo-finite fields together. ¢ O ¦üy8mb zmQPuq| n| rV~¥y8r¢pAnlnm q|q| C|4mbz R m<~n Clq4!SDTU ÌSv{r6z0VÐ| y¥QnzrVqt §| y V z}| | zqmI~WUYXSKZ ~¥nr¢CFC| rVX| ~"nlF&¨m2wQAzqr¦v2rQ~"pSmbV qt4p¥4pn| Clqv{mbC| rQogmIpn4n| rV~ùÌv{r6z0VÐ"k"lnm We can be precise about the way in which the counting measure bundles nlnm \[íÌv{r6z0VÐ"| ~"nlnm¢t mw4~"ogrQ~n| C| m¢|bSmbQmbp^]¥y8rQpAl| ¢lW[ _ Í Ìv{r6zT 0VÐS the counting measures #q(φ). Each formula φ gives a function q 7→ #q(φ), ´Qº¯®bº(r¢y an integer-valued function on the set of prime powers. Let F be the ring of all integer-valued functions on the set {pr} of prime powers. Declare two functions equivalent, if they take the same value at pr for all r and for all but finitely many p. Write F/ ∼ for the quotient of F under this equivalence relation. 1We have a moving lemma: the congruence relation on the scissor group can be used to relabel the free variables of a formula, so that free variables of the two factors are always distinct. WHAT IS MOTIVIC MEASURE? 7 P È ¦ Ó ¦ " £m £ Theorem 2.4. There exists a ring homomorphism N from the scissor ring ÖF×ÜCàÒå;Û2Übæe北îMâBî¢åçï;ïâ6öuëÒïçæeå;× ÜCàÒâå;á2ÜCâ6ÛbÜ]ßqî<å;ÜCà è6éeêëÒï;â6ÜbâBëÒáCéuéeþ.Û]ßVæe×ÒäëÒÙÒÜ2ÜCàÒâBábâ6ÛCÙÒï;Ü2æe×ÒäåçäÒâ6æeÛ Sring to F/ ∼ that respects counting: #∗(φ) = N([φ]). ÿ ÿ å;× æeëÒëÒáCéeëÒáCåçæeÜCâ àÒåçÛCÜCéeáCåçè6æeï&è6ée×ÜCâ6öuÜ â6ÜCæeåçï;Û<î¢åçï;ï â â6ïçæ éeáCæeÜCâäãée×ãåç×ãæþ.éeáCÜCàÒè6éeêå;×ÒÚÄæeáCÜbå;è6ïçâ ¢; ¢ In other words, with only a finite amount of ambiguity, the counting ( ( S "M¡ £¢ å;Û¥ÜC驦Ò×Òäæ ø¥¤uÙÒåç褣uûþ.éuéeï;ëÒábéuéeþ¨æeïçÚeéeáCå;ÜbàÒê ÜCéäÒâÜCâ6áCêå;×Òâ>î<àÒâ6ÜCàÒâáæ Úeå¡ eâ6× ÿ eâ6áCóeée×ÒâS£§×Òéqî<Û¨§ª©y«¥¬&¯®«±°^«¥²5«¡³&´ß(î<àÒâ6×ÃîMâÜCábó ÜCéäÒå¥ uåçäÒâµ: óãâM eâáCóãåçקÜCâÚeâ6á å;×ÜCâ6ÚeâáXå;ÛXëÒábå;êâ measure specializes to counting solutions to ring formulas over finite fields. ¢g¦ ÿ ÿ åç× ÜCàÒâáCæe×ÒÚeâÝ r r : àÒâ קÙÒê â6áéeþAÛCÜbâ6ëÒÛåç× ÜbàÒå;Ûjæeï;ÚeéeáCåçÜCàÒê î<å;ïçï â æeܦï;âæeÛCÜjÜCàÒâ ¶ ¶ To say that N is a ring homomorphism is to say that it is compatible with ¢n¨ ÿ ×ÙÒê â6áXéeþå;×ÜCâ6Úeâ6ábÛ îMâè6ée×ÒÛCåçäÒâ6á]ßÒî¢àÒåçèQàãå;Û<ÛCéeêâ6ÜbàÒå;×ÒÚïç奣eâS :Gßuåç×ãÜCàÒâîMéeáCÛCÜ<è6æeÛCâk#Sî¢àÒâ×k: ¶ ∼ products and Fubini. Unlike the earlier isomorphisms for polygons Spoly = R [ ÿ : åçÛMáCéeÙÒÚeàÒïçóÄÝ ·¤¸ î<àÒâ6áC⩹ å;ÛBÜCàÒâ¦×ÙÒê â6áGéeþ"äÒå;ÚeåçÜCÛBéeþ:íî¢àÒâ× î<áCå;ÜbÜCâ6× W' ð¦éeÜCâXÜCàÒæeÜo å;ÛBëÒábå;êâ ∼ ¢ and finite sets Scount = Z, here we make no claim of isomorphism between ÿ å;× å;×ÒæeáCó$#Sæe×Ò亹å;Û<áCéeÙÒÚeàÒï;ó$#SïçéeÚ: 'h»#Tï;éeÚ Ý&'5' ¢ the scissor group Sring and the target ring F/ ∼. ÿ àÒâ¼³½¾W¿MÀW§ª«±°&¿Xåç× ÜCàÒåçÛjæeáCâ6æ àÒæeÛ â6â6× ÜCé è6éeêâÙÒë î¢åçÜCàÃæe× æeïçÚeéeáCå;ÜbàÒê î<àÒå;èàîMéeáy£uÛ¦å;× ×Òé ¨ qÁW¹ÂÛCÜCâëÒÛå;× ÜCàÒâAîéeáCÛCÜèæeÛCâ/ßî<àÒâ6áCâgÁ2æe×Òäqà æeáCâ¥ÛCéeê⯦Òöuâ6äë¨éeÛCåçÜCå¥ eâ¥è6ée×ÒÛCÜbæeקÜCÛ-ÄVÜCàÒæeÜ êéeábâAÜCàÒæe× The proof of the theorem relies on ultraproducts, a standard tool in logic. ÿÒÿ å;Ûwßuæe×ãæeï;Úeéeábå;ÜCàÒê î<àÒå;èàîMéeáy£uÛGå;× Å³&¥ÆK´t³&Çd«¥¬&;È«¥ÇÉ¿©#Tî¢àÒåçèQà å;Û<éeþ.ÜCâ6× æ áCâM uåçæeÜCâ6ä æeÛËÊK' åçÜCà ¢Ì ÿ æe× æeï;ÚeéeáCåçÜCàÒê ée×ÒâÄâ6ö§ë¨â6è6ÜbÛÜCàÒæeÜ ée×Òâè6æe× áCæeëÒå;äÒïçó äÒâÜCâ6áCêå;×ÒâÄî<àÒâ6ÜCàÒâ6á æe×óùøáCâ6æeÛCée×Òæ ïçó ÛCÙÒèà 2.3. Improving the scissor ring. The shortcoming of the scissor ring ¡ 6â6äÒûÄåçקÜCâÚeâ6á¢åçÛ>ëÒáCåçêâ ÛCå S is that is too much about it has been left inexplicit. In our discussion ¢ ring ÿ Gâ6þ.éeáCâ ÜbàÒâîMéeá5£ éeþ2ئÚeáCæqîæeï4ߪòæóeæeïæe×ÒäõÒæeöuâ×ÒæúÜCàÒâ þ.æeÛCÜCâ6Ûbܦæeï;ÚeéeáCåçÜCàÒê îéeá5£eâ6äå;×æ éeÙÒÜ Í of the area of planar polygons, we found a handy set of generators (unit ÿ ¹"Î Ï8Ð{Î Ï8Ð ÛCÜCâ6ëÒÛ àÒâ6å;áBæeïçÚeéeáCå;ÜCàÒê îMéeá5£§ÛBå;×úæ éeÙÒÜo¹KÑ\ÒÔÓBÛCÜCâ6ëÒÛE#Sæe×ÒäúÜCàÙÒÛÄøÖÕb¿M°&¿R´tת¬&´b®×J¬&×JØt¬&¡Ùm¡³&Ú&² · ¢¨ width rectangles). Our current aim is to find a handy set of generators of ÿ ÿ ²^ÛÜÉÀM¿bû&'^ÄÒæe×ÒäãæêéuäÒ塦Òè6æeÜCåçée× ó6Ý$â6×ÒÛCÜCábæÄæe×Òä*Þ2éeêâ6áCæe×Òè6âå;×æ éeÙÒܵ¹"ßXÛbÜCâ6ëÒÛ ¢ a somewhat modified scissor ring Smot. The idea is to take a ring formula, ÿ (ªà( KP ªáP =â § ªã} "ä-(§Ø¦áCâÜCàÒâ6áCâîMæóuÛ<ÜCéábâ6è6éeÚe×Òå¥ â ëÒáCåçêâ6Û¢éeÜCàÒâ6á<ÜCàÒæe× ó ÜbáCå;æeï&äÒå¡ uå;ý and through a process of “quantifier elimination” arrive at an equivalent ÿ tåçæ×ÒâîMæó ÜCàÒæeÜ<è6éeêâ6Û¢ÜCé êå;×Òäãå;Û ó ÙÒÛCåç×ÒÚ ÛCåçée× ring formula that does not involve any quantifiers (that is, the symbols ∀, ∃ L L è ä- é䡨ê K "ã #5Wë&ëeÞ&'h(6ìy47839."35,Ë:=<¡Ý602IgF,h02]*3µ0 D-4mV +M4?jik0 :ZVM021^0yVP35IS#¥:6W'^í\vî"a will be eliminated). A formula without quantifiers belongs less to the realm of logic than to the realm of algebraic geometry. A suggestive example of a ÿ ÿ àÒâ ëÒáCé ï;âê àÒâ6áCâ ÜCàÒéeÙÒÚeàåçÛXÜCàÒæeÜXÜbàÒâ6áCâ å;Û>×Òéúé §å;éeÙÒÛ¦îMæóãÜCéè6éeêëÒÙÒÜCâ #¡:W'^íªábæeëÒå;äÒï;ó ¨ quantifier-free formula is #Séeá<âM eâ6×ï#¡:*W'-ío#Sêéuäd: '5' ئ×ÒéeÜCàÒâ6á<å;äÒâ6æå;Û¢ÜCéÙÒÛbâ ¢ (f1 = 0) ∧ (f2 = 0) ∧ · · · ∧ (fn = 0). Z ¡ ä- ^¢! ÖðPéäGâ!j! ã} ªm#yWñ&ëeÞ&'h(|cO35,M335ò^0I¤7ªID FH+M?ji4m+M]q0yDM?Pó #¥u ôW~yu ~-õ-õ-õ\~yu 'ö ÷ u ôM~yu ~-õ-õ-õ\~yu [ [8ß [ [8ß8ø L L ' N;O35,M3 ~-õ-õ-õ\~ + V¡39.^,W3h3UùúM>©N;027JOï7ªOA3EF,W+yFH35,h7Ji=7ªOADM7E7JO3kI357k+ FH+MI¤07ª021K3U1"DM?j_35I ó # ôM~ ¶ ¶ ¶ [8ß [ That is, the zero set of an affine variety. In fact, we will find that the L DM,M3qDM??m75DMûK354º78+ B¤3¯F+MI¤027J021"3/0247539.K35,hI¤>0In7ªOA3µIDM]*3qDMIn7JO3µI357`+ F,h0]635Ia ôM~-õ-õ-õ\~ ¶ ¶ [8ß improved scissor ring is defined as a quotient of the free abelian group on ü¤§ÙÒå;褣uïçó åçäÒâ6×ÜCå;þ.óíî<àÒâ6ÜCàÒâá æÚeå¥ eâ6× åçקÜCâÚeâ6á å;Ûæ æeï;ÙÒâéeþ`ó{ß âúêå;ÚeàÜàÒéeë¨âúÜbéÛCéeêâ6àÒéî the set of varieties over Q. The details of this construction will reveal what Ì ÿ ¦ÒÚeÙÒáCâ6äéeÙÒÜ<àÒéqî ÙÒÜ>×Òéée×Òâ àÒæeÛ<óeâ6Ü is so motivic about motivic measure. ¢ àÒâ6áCâMæeábâêæe×óëÒï;æeè6â6ÛÜCàÒæeÜëÒábå;êâÛ{è6éeêâAÙÒëå;× ÜCàÒâêæeÜCàÒâ6êæeÜbå;è6æeïï;åçÜCâ6áCæeÜCÙÒábâ/ßæe×Òäêæeקóéeþ ¨ mý ÛCéeêâXéeþ&ÙÒÛMî<àÒé æeáCâXåçקÜCâáCâ6ÛCÜCâä§åç×úëÒáCå;êæeï;åçÜó ÜCàÒâ6Ûbâ¦êå;ÚeàÜÛbÙÒÚeÚeâ6ÛCÜBæ îMæóÜbé å;äÒâ6×ÜCå;þ.óëÒáCå;êâ6Û 2.4. A scissor ring for coverings. Each ring homomorphism f : Sring → ÜCâ6ÛbÜCå;×ÒÚ æeï;îæqó§ÛXï;é§é&£ æeܦæe×óuÜCàÒåç×ÒÚú×Òâ6îÜbàÒæeܦîMâ ï;â6æeáb× î<å;Übàée×Òâ â6óeâ éeë¨â6× ÜCéúÜCàÒåçÛ¦æeëÒëÒï;åçè6æeÜCå;ée× ¢ R defines a specialization of the counting measure þ¦éîMâM eâ6á]ßþ.éeá ÜbàÒâúáCâ6êæeåç×ÒäÒâ6á éeþBÜCàÒåçÛ¨¦ÒáCÛCÜ àÒæeï;þBéeþGê ó ÜCæe北 Öîæe×§Ü ÜCéãþ.éuè6ÙÒÛée× ée×Òâúè6ïçæeÛCÛCå;èæeï æeëÒëÒáCéeæeèà φ 7→ [φ] → f[φ] ∈ R. ¢ (ªÿ(ãE ã} ={ ãç¯ " äkꧢ çã}î t¡ " "äy¡ ª }â â! K 8¡ ª "ä-( æ×Òâ éeþ2ÜCàÒâêéeÛbÜ>æeêæ& å;×ÒÚ The ring F/ ∼ is one of many possible specializations R. £u×Òéî<×æeÛ å;Û Another specialization of Sring comes from n-sheeted covers: L L x K §ã}&¡é䢡G £-¡ 㡨ê K "ã #y¤£¦¥&ë&'h(*ì : 0I¼DoF,h0]63/7ªOA354k: VM021^0yVP35I`w w +-,dDM??0247839."38,hI Definition 2.5. We say that one formula φ(x) is an n-sheeted cover of an- wa other formula φ0(x0) if there exists a ring formula ψ(x, x0) such that for every § " eâáCÛCâ6ïçóßuå;þ$å;×ÜCâ6Úeâ6á!:f®³¿R²Ë´t³§g®«±°^«±®t¿ow&xn wþ.éeá<Ûbéeêâjå;×ÜCâ6Úeâ6á/wªßuÜCàÒâ6×d:ôåçÛBè6éeêë©éeÛCå;ÜCâ ée× pseudo-finite field of characteristic zero, ψ gives an n to 1 correspondence ¢ between the solutions x of φ(x) and the solutions x0 of φ(x0). Example 2.6. Let φ(x) be the formula ‘x 6= 0’ and let φ0(y) be the formula ‘∃ z. (z2 = y) ∧ (y 6= 0)’. The formula ψ(x, y) given by ‘x2 = y’, presents φ as a 2-sheeted cover of φ0. 8 THOMAS C. HALES The congruence condition for Sring asserts that if φ is a 1-sheeted cover of 0 φ , then they give the same class in Sring. A broader congruence condition can be given as follows. 0 ¢£ ¤¦¥§£©¨ ¡ ¤¦¡ ¤¦§ ¤ ¤¦¡ ¤¦ Congruence (Covers). If φ is an n-sheeted cover of φ for some n, then ¢¡ ¤¦ ¡ ¤¦¤¦ ¢£ ¤ [φ] = n[φ0]. ¥ We may form a new scissor ring Scover with this broader congruence ) #&*$ "+",.-/-/% condition and the old scissor relation. We have a canonical surjection "!$#&%(' S → S . N©143©OQP"R HS=6DT36UVHSDF581CWV3X9WV3>74=RYRZ1496[C\]1CK ring cover 021436587496:;143<:;=>:;?61A@B1C@B=6DFEG=IH¦J ^A_V`baVcedVfga(hjiFk"lnm¦oqpSr¢sut mwvxrQy{z}| ~n|V q| ~nl| oqpC|v{mw qv$sgmbpS~ByYpSrgvbrVv{ogr¢~C| m w vsgmIp~CQz |n nlnm t4mIp|br Clm6| pBopn|v{mXyS4brQp~ | ~Gnrnrsgmjrgnm¦rQy2nlnm v2rQ~|v{ogrQpC¢b 2.5. The scissor group of motives. Generators. Let VarQ be the cat- rQy{pmI~rVt Qz ~SmbyS t| FpC| nlv{mbn| w jlq4~Mm6VFQmbznlnm |nzu ~Sp§QnzA| ~zrVvrQyMQnw| m6bXQzv{r6zqmIpn {rV qt z sgmG~n nogmbpS} nrg n~ r¦zu| ~6 ~S~2nlnmBoqpSr¢sut m6v¡4¢t mwn¢Clq egory of varieties over the field of rational numbers Q. We take the free QmbrVv{mbmIp~2SrX~n n¢l¢>mIwm6bnl4>| mbmIpClm6t mb~S~{mvM n~2brVnyYmI~~ nl4>Qtt(v{mbClr6z~ nlF¢lmBsgmImw>oprQogrQ~mbzClw ~2yS4p>4pmGm6| nlnmbp abelian group generated by the objects of VarQ. £ ImIzrjmIp~ogmIw|Qtgw4~mI~ArQpX4pmG~SrtFsgrQpn| rg n~XQnz zu| ¤Gw t {Clq4¥mbm6¢y8r¢p¢w vsgmIp~ nlF 1 pSmb~Spn| S~rQyn4sut mb~MIrgn~Spn nbSmbzswmI~n|v<4sut m v{mw}nlnmbpSnlnm>ouFC| mwnbmXrQy An example of a element of the free abelian group is [A ], the genera- zr nr¢BmIwbmbmIznlnmt|vG| bmIz¢6¢t 6 qtFSr¢pC§¦BnzXnlnmb~SmGv{mbClr6z~zrjnr¢ 4oo}t 4 QttrXtFpS¢mIp w vsgmbpS~ ¨ tor attached to the affine line. This particular generator will be of special mImw2nlnm{opn4bC| ¦lq4oqoVmwn~&nlFClmpnQ|nmbz¢wQt w t4rQp¨A|tt¢sVm{~n n¤Gw| m6bnt MpSmb 4pzmbz 1 y8pmI© m6bnt pªySFIrQp~~SrAClq42| ©A|ttnbrVv2oVmwn~n4my8rQp¨Clm$n|v{m$~ogm6bn&«q pSnlnmbpn¬®¯° ±V²q°¬³j´Qµ importance in the constructions that follow. We write L = [A ] for this w vsgmIp~ªr¥nlnmw| element and for its image in various scissor groups. (The ‘L’ is for Lefschetz, ¬Sq®2¶S·b°®I²q·I®{°¬¶S®b¸µ¶®I®b¹G¶&¬´º®b»¼n°ºS®{¬½¢¬&®I¾®IºY³G¿(´Q¶¶°.À¸®2¹G®I½¢²¶¥À¢®2®bÁb¿u¸´Qº®I¯Xµ8´¢º&¬Sq®2¶S´¢¸Â¼n¬°´¢²X´Qµ}½ ~"|&nlnm FC pSmrQyqClmoprQsut mwvÄnl4½¢²b³>v{mInlnr6zA|tt as in Lefschetz motive.) ¿uº´VÀ¸®b¹Ã¶´{®I¸®T±V½Q²w¬u½Q²q¯A¶´{·I®b¸®6ÀºS½¢¬S®b¯A©"| sgmIbrVv{m>v{rQpm brVv{o}t| w4mIzF~$nlnm¢w qv$sgmbpS~$QmbAt4pQmbpC mb¢mbpClm6t mb~S~n/|Clm y8rgtt rnA|v{mInlnr6zq~ £ ¤Gw t n| mI~<|nbpm6F~SmBpn4nlnmbp¥~nt rnAt Äjk"lmGSmb¢lq| © nmb~2Clq4A¨mbpmBopmI| rV ~Ct nrnA>rg t z There are two types of relations: scissor relations and congruence rela- ClmGzu| pm<|bSrgt mbpn4sut mGt4sgrQpmImw¥yYrQp"nlnm¢v2rQ~|nzqmIySFC| g4sut m2wQt w t4rQpn Å tions. Our scissor relation will be rather crude, but justifiably so, since the pSmb© | y8pSrgvÆ4pC| wt m<Ç¢ÈQÉ{r¢y&Ê2°¶S»¼°¶°¬S°´¢²®b¶&˺°¬Sq¹G®I¬°·I½¢® ÌSÍ¢ÎQÏQÍ¢Ð(swÑ"«Ò§Ó¦AÔÕÕq Zariski topology is a coarse topology that limits the possibilities for a scissor relation. The only cutting that will be permitted is that of partitioning a variety into a closed subvariety and its complement. Scissor Relation. If Z is a closed subvariety of X, then ÖF×ÆئÙÒÚeÙÒÛCÜÝeÞeÞeÝªß ÜbàÒáCâ6âãÖF×ÒäÒåçæe×Æè6éeêëÒÙÒÜbâ6áÛCèå;â6×ÜCå;ÛbÜCÛ]ßAìÃæe×Òå;×ÒäÒábæíئÚeáCæîMæeï4ß ð¦â6â6áCægñ òæóeæeï æe×Òäôð¦å;ÜCåç×ôõÒæeö§â6×Òæªß{è6ée×ÒÛCÜCábÙÒè6ÜCâ6ä÷æùøë©éeï;ó§×Òéeêå;æeïAÜCå;êâúëÒáCå;êæeï;åçÜóÜCâ6ÛCÜCûªß{æÃê ÙÒèàüÛCéeÙÒÚeàÜCýSæeþ.ÜCâ6á ÿ ÿ ÙÒÜâï;ÙÒÛCå¡ eâ>Úeéeæeï(éeþ¨áCâÛCâ6æeáCèàÒâ6áCÛBåç×ÄÜCàÒâ¢æeï;ÚeéeáCåçÜCàÒêåçè<×ÙÒê âáMÜCàÒâ6éeáCó îMéeáCïçä ìÃéeÛCÜÛCàÒé§è¤£uå;×ÒÚ îMæeÛ [X] = [Z] + [X \ Z]. ¢ ÜCàÒâXÛbå;êëÒïçå;è6åçÜFó æe×Òäúéeábå;Úeå;×Òæeïçå;ÜFóéeþÜCàÒâ6åçáMÜCâ6ÛbÜ î¢àÒâáCâ6æeÛBÜCàÒâ øâöuë¨âáCÜCÛCûàÒæeäúêæeäÒâXè6éeêëÒï;å;èæeÜCâ6ä ¢¥¢¡¢ êé§äÒ奦Òè6æeÜbå;ée×ÒÛ2ée× â6ö§å;ÛCÜCåç×ÒÚ¦ÜCâ6ÛCÜbÛ2ÜCéjÚeæeå;× å;êëÒáCé§ eâ6êâ6×ÜCÛ]ßgÜCàÒâÛCâBæeÙÒÜCàÒéeáCÛ2áCâÜCàÒéeÙÒÚeà§Ü ÜCàÒâBäÒåçáCâ6è6Übå;ée× ÿ å;× î<àÒå;èàÜCé ëÒÙÒÛCà ÜCàÒâÙÒÛCÙÒæeï å;äÒâ6æeÛjî<å;ÜCàÃÛCÜbÙÒ×Ò×Òå;×ÒÚãÛbÙÒè6è6â6ÛCÛ àÒâ6åçájæeï;ÚeéeáCåçÜCàÒê å;Û æeÛCâäée× ÜCàÒâ ¢©¨ ÿ ¥ 6æeÜCåçée×ãéeþ{ëÒábå;êâ×ÙÒê â6áCÛ The congruence relation is more involved than the scissor relation. If we þ.éeï;ïçéqî<å;×ÒÚâ6ï;âÚeæeקÜ<èàÒæeáCæeè6ÜCâ6ábå make a direct translation of the congruence relation for the scissor group of ¢ §§ § ÷£ ! " $#SÝeÞeÞ&%&'¤(*)+-,/.§021"35460247839."35,;:=<¡Ý">?357A@CB¤3EDGFH+-I¤027J021"3G0247539.K35, ring formulas, we might guess that the congruence condition between two ¥ L L L +M,EN;OP0RQ8OS:TODMIU+M,WVP38,YXZ#SïçéeÚ: '\[S]6+^VM_K?+`@badcOA354e:f02IgF,h02]*3`0 DM4V +M4?jik0 varieties X and Y should be the existence of a correspondence Ψ between X L :f02IY4m+M7nDoFH35, 3¤Q57bFH+MN/35,h> l L DPQ578+M,Yrs@t> and Y that induces a bijection between X(K) and Y (K) for every pseudo- :pVP+§35In4m+M7GOD-1"3qDM4iEF,\02]63 l L @ïçéeÚ:/a #¥udvTw&'yxkzsu{xEvfw|#Sêé§ä}#¥:!~yuW'5' +M,U3hDPQ5O 0247539.K35,YwK~ rwqr finite field of characteristic zero. This first guess is suggestive: the congru- l ence relation should involve an algebraic correspondence. This suggestion k§woVmb~mI"sw/& k¤ lands us deep in the territory of motives. Here is the precise definition of Í the congruence relation. Congruence Relation. [X] = [Y ] whenever X and Y are nonsingular projective varieties that give the same virtual Chow motive. We will uncoil this definition a bit below. All that is ‘motivic’ about motivic measure stems from this particular congruence relation. 20 THOMAS C. HALES WHAT IS MOTIVIC MEASURE? 9 [14] J. Pas, Uniform p-adic cell decomposition and local zeta functions, J. reine angew. Definition 2.7. The quotient of the free abelian group by the scissor and Math. 399 (1989), 137–172. congruence relations is the motivic scissor ring K. (The letter ‘K’ is the [15] J. Sebag, Int´egrationmotivique sur les sch´emasformels, preprint. standard notation for a Grothendieck group, which for our purposes is just [16] Scholl, Classical Motives, in Motives, U. Jannsen, S. Kleiman, J.-P. Serre Ed., Proc. another name for a scissor group.) The localized version K[L−1] ⊗ Q will be Symp. Pure Math., Vol 55 Part 1 (1994), 163-187. called the localized motivic scissor ring and denoted Smot. (It will become clear in Section 3.6.3 why it is useful to invert L.) It is time to uncoil the definition of this congruence relation. There is a category of Chow motives. To describe this category, we assume familiarity with the Chow groups Ai(X) of a variety X. They are groups of cycles of a given codimension i modulo the subgroup of cycles that are rationally equivalent to 0. A detailed treatment of cycles, rational equivalence, and Chow groups can be found in [8]. Other good treatments of Chow motives can be found in [16] and [9]. An object in the category of Chow motives is a triple (X, p, m) where X is a smooth projective variety of dimension d, p is an element in the Chow ring Ad(X × X) that is a projector (p2 = p), and m in an integer. The set of morphisms from (X, p, m) to (X, p0, m0) is defined to be the set p0Ad+n−m(X × Y )p. Varieties that are not isomorphic as varieties can very well become isomor- phic when viewed as Chow motives. For example, isogenous elliptic curves are isomorphic as Chow motives. There is a canonical morphism from the Grothendieck ring of the category VarQ to the Grothendieck ring of the category of Chow motives. We let K be the image of this morphism. To say that two varieties are equal as virtual Chow motives is to say that they have the same class in K. 2.6. The motivic counting measure. The following theorem follows from a deep investigation of Chow motives, and the theory of quantifier elimina- tion for pseudo-finite fields. Theorem 2.8. There exists a unique ring homomorphism Scover → Smot that satisfies the following property (Zero Sets). Zero Sets. If φ is a ring formula that is given by the conjunction of poly- nomial equations, then [φ] is sent to the affine variety defined by those polynomial equations. There are ring homomorphisms Scount → Scover → Smot. We use the notation φ 7→ [φ] for the class of φ in any of these rings, depending on the context. Version: Dec 8, 2003. Work supported by the NSF Definition 2.9. The composite map φ 7→ [φ] ∈ Smot will be called the Copyright (c) 2003, Thomas C. Hales. motivic counting measure of the formula φ. This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/ or send a letter to The motivic counting measure of a ring formula is thus represented by a Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. rational linear combination of varieties over Q. I like to think of the motivic 10 THOMAS C. HALES WHAT IS MOTIVIC MEASURE? 19 counting measure as counting the number of solutions of the ring formula 4.5. Concluding Remarks. This article is an exposition of a particular over finite fields in a way that does not depend on the finite field. Instead version of motivic integration, called arithmetic motivic integration. Proofs of giving the answer as a particular number, it gives the answer in terms of of results stated in this article can be found in [6] and [4]. Motivic integra- a formal combination of varieties having the same number of solutions over tion has been developing at a break-neck pace, ever since Kontsevich gave a finite field. Here is the precise statement. the first lecture on the topic in 1995. The version of motivic integration P developed in the late nineties goes by the name of geometric motivic inte- Theorem 2.10. Let φ be a ring formula, and let a [X ] be a represen- i i gration. Geometric motivic integration is a coarser theory, but is sufficient tative of the motivic counting measure [φ] as a formal linear combination for many applications. Good introduction are [1] and [12]. Some articles on of varieties. Choose a model of each X over Z. For all r and for all but i geometric motivic integration include [3] and [7]. Another version of motivic finitely many primes p, the number of solutions of φ in Fpr is equal to X integration has been developed by J. Sebag for formal schemes [15]. See also ai#Xi(Fpr ). [11]. Cluckers and Loeser are in the final stages of preparation of an ultimate version of motivic integration that subsumes both geometric and arithmetic Example 2.11. As an example, let us calculate the motivic counting mea- motivic integration [2]. sure of the ‘set’ of nonzero cubes. The formula is given by We began this article by stating that motivic measure does not fit neatly φ(x):‘∃y. (y3 = x) ∧ (x 6= 0)’. into the tradition of Hausdorff, Haar, and Lebesgue. However, a major result states that the motivic measure specializes to the additive Haar measure on The scissor relation can be used to break φ into two disjoint pieces φ = locally compact fields (Theorem 3.10). Thus, the motivic measure is perhaps φ ∨ φ : the part φ on which −3 is a square and the part φ on which it 1 2 1 2 not so peculiar after all. In fact, in many respects it is strikingly similar is not. Let M be the class in S corresponding to the zero-dimensional mot to the additive Haar measure on locally compact fields. It has been my variety x2 + 3 = 0. The class M has two solutions or no solutions according experience when I calculate motivic volumes to lose track – mid-calculation as −3 is a square or not. When −3 is a square, the cube roots of unity lie – of which measure is being used. in the field, so that the nonzero points on the affine line give a 3-fold cover 3 of φ1 (under y 7→ y ). Thus, φ1 has measure µ ¶ L − 1 M . References 3 2 [1] A. Craw, An introduction to motivic integration, 1999. On the other hand, if −3 is not a square, each non-zero element of a pseudo- [2] R. Cluckers and F. Loeser, Fonctions constructibles et int´egration motivique I, II, in finite field of characteristic zero is a cube, so that φ2 has measure preparation. µ ¶ M [3] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic (L − 1) 1 − . integration, Inventiones Mathematicae 135, 201-232 (1999). 2 [4] J. Denef and F. Loeser, Definable sets, motives and p-adic integrals, JAMS, 14, No. 2, The sum of these two terms is the measure of φ in Smot. 429–469, 2000. [5] J. Denef and F. Loeser, On some rational generating series occuring in arithmetic 3. Locally Compact Fields and Haar Measures geometry, to appear. [6] J. Denef and F. Loeser, Motivic integration and the Grothendieck group of pseudo- This section makes the transition from finite fields to locally compact finite fields, Proc. ICM, Vol II (Beijing, 2002), 13-23, Higher Education Press 2002. fields and from counting measures to additive Haar measures. [7] J. Denef and F. Loeser, Motivic Igusa functions, Journal of Algebraic Geometry 7, In Section 2, we developed a universal counting measure for ring formula. 505-537 (1998). It may be viewed as counting solutions to the ring formula over a finite field [8] W. Fulton, Intersection Theory, second edition, Springer, 1998. [9] G. van der Geer and B. Moonen, Abelian Varieties, preliminary version of Chapter XI. in a way that does not depend on the finite field. The Fourier transform and the Chow ring, July 2003, Counting measures are a rather simple and uninteresting type of measure. http://turing.wins.uva.nl/bmoonen/boek/BookAV.html.~ In this section, we construct a universal (motivic) measure with ties to locally [10] M. Kontsevich, lecture at Orsay, Dec 1995. compact fields. This new measure may be viewed as the volume expressed in [11] F. Loeser and J. Sebag, Motivic integration on smooth rigid varieties and invariants a way that does not depend on the locally compact field. To carry out the of degenerations. Duke Mathematical Journal, 119, 315-344 (2003). [12] E. Looijenga, Motivic measures. S´eminaireBourbaki, Vol. 1999/2000. Ast´erisqueNo. construction, we must work with a different collection of formulas (called 276 (2002), 267–297. DVR formulas) that are better adapted to locally compact fields. ‘DVR’ is [13] E. Lupercio and M. Poddar, The Global McKay-Ruan correspondence via motivic an acronym for discrete valuation ring. integration, preprint, 2002. 18 THOMAS C. HALES WHAT IS MOTIVIC MEASURE? 11 Euler characteristics and Hodge polynomials! For example, the formula for 3.1. Examples of rings. To make the transition from finite fields to locally the nonzero squares in a field compact fields, we wish to replace ring formulas with formulas in a language that has a rich assortment of locally compact structures. ‘∃y. (y2 = x) ∧ (x 6= 0)’ Example 3.1. Let C[[t]] be the ring of formal power series with complex has Euler characteristic zero. coefficients. A typical element of this ring has the form X∞ 4.2. Geometry of varieties. There is a motivic change-of-variables for- i mula that is similar to the standard change of variables formula in calculus. x = ait Using this formula, it is sometimes possible to show that two birationally i=k equivalent varieties have the same motivic volume. This has deep implica- (with no constraints on the convergence of the series). Pick the initial index tions for the geometry of the two varieties. In particular, the motivic volume k so that ak 6= 0 (if x 6= 0). determines the Hodge polynomial of the varieties. The valuation of x is defined to be the integer k: This approach was followed by Kontsevich, who used a change-of-variables val(x) = k. calculation to show that birationally equivalent projective Calabi-Yau man- ifolds have the same Hodge numbers [10]. Applications to orbifolds appear The angular component of x is defined to be the complex number ak. × in [13]. ac(x) = ak ∈ C . (In the special case x = 0, we set val(0) = ∞ and ac(0) = 0.) 4.3. Computation of p-adic integrals. Many integrals over p-adic fields are notoriously difficult to calculate. Motivic measure exposes the underly- The name angular component is not meant to suggest any precise con- ing similarities between volumes on different p-adic fields. It gives a deci- nection to angles. The name is based on a loose analogy with the polar sion procedure to calculate p-adic integrals (at least when the data defining coordinate representation of a complex number: just as the angular com- the integral can be expressed as DVR formulas that can be reproduced at ponent θ of a nonzero complex number reiθ distinguishes among complex some finite level m). In particular, this means that a computer can be numbers of the same magnitude (or valuation) r, so the angular component programmed to compute a large class of p-adic integrals. of a formal power series helps to distinguish among formal power series of a given valuation k. 4.4. Generating Functions. Motivic counting gives a way of counting There are many other rings with similar functions, ac and val. For ex- that is independent of the finite field. Let ample, we can change the coefficient ring of the formal power series from C X∞ to any other field k to obtain k[[t]]. Or we can take the field of fractions of (p) i Zp(t) = ai t k[[t]], which is the field of formal Laurent series with coefficients in k: i=0 X∞ k((t)) = { a ti | a ∈ K}. be a generating function, where the constants a(p) are obtained by counting i i i −N solutions to a formula in some p-dependent way. (Each generating function depends on a single prime p.) Motivic measure can often give a way of For each prime p, there are valuation and angular component functions forming a p-independent series defined on the field of rational numbers. If x is a nonzero rational number, pick integers a, b, c, N so that X∞ Z (t) = [a ]ti bpN+1 mot i x = apN + , i=0 c where c is not divisible by p, and a ∈ {1, . . . , p − 1}. The integers a and N taking values in S*[[t]] and specializing for almost all p to the p-dependent series Zp(t). The motivic series collects the behavior of the various series are uniquely determined by this condition. Define the valuation of x to be Zp(t) into a single series. valp(x) = N ∈ Z and the angular component of x to be image of a modulo Denef and Loeser have studied motivic versions of Hasse-Weil series, Igusa p in Fp. series, and Serre series. They have used the general motivic series to prove Example 3.2. If p = 2 and x = 17/8, then that various properties of these series are independent of the prime p. See −3 [5]. 17/8 = 1.2 + 2, val2(17/8) = −3, ac(17/8) = 1 ∈ F2. 12 THOMAS C. HALES WHAT IS MOTIVIC MEASURE? 17 Other examples, can be obtained from this one by completion. For each 3.7. The universal nature of motivic measure. Just as the motivic p, counting measure counts solutions to ring formulas over finite fields in a d(x, y) = (1/2)valp(x−y) field independent way, so the motivic measure takes the volume of a DVR 2 is a metric on the set of rational numbers. The completion is a locally formula over locally compact fields in a field independent way. There is a good theory of measure on locally compact fields. This is the compact field, called the field of p-adic numbers Qp. The valuation valp and angular component function ac functions extend to the completion. Haar measure, which is translation invariant. Given a DVR formula φ and a locally compact structure K with ring of integers OK , we can take the 3.2. The DVR language. We have seen by example that there are many volume of the set of solutions to the DVR formula rings with functions val and ac. In each case, there are three separate rings (11) vol({x ∈ On | φK (x)}, dx). that come into play: the domain of the functions val and ac, the range of the K function val (which we augment with a special symbol {∞} for the valuation The measure dx can be given a canonical normalization by requiring that it n of 0), and the range of the function ac. We call these rings the valued ring, assigns volume 1 to the full set OK . the value group, and the residue field, respectively. We are now ready to state the main result on motivic measure. Like all We formalize this relationship as a language in first-order logic with func- the other principal results in this article, the result is due to J. Denef and tion symbols val and ac. We allow ourselves to build syntactically well- F. Loeser. formed expressions with variables, parentheses, quantifiers, the function Theorem 3.10. The motivic volume of φ is universal in the following sense. P −N symbols val and ac, the usual ring operations (0, 1, (+), (−), (∗), (=)) on Let ai[Xi]L i be any representative of the motivic volume of φ as a the valued ring and residue field, and the usual group operations and in- convergent formal sum of varieties over Q. Pick models for the varieties over equalities on the value group (0, (+), (≤)). These formulas will use variables Z. After discarding finitely many primes, for any locally compact structure of three different types xi for the value ring, mi for the value group, and ξi of the DVR language, the K-volume of the formula is given by a convergent for the residue field. Quantifiers ∀, ∃ can be used to bind all three sorts of sum (in R) X variables. a #X(F )q−Ni , The construction of first-order languages is commonplace in logic, but i q even without any background in logic, it is not hard to guess whether a where Fq is the residue field of K. formula is syntactically correct. We allow standard mathematical abbrevia- This wonderful result states that the Haar measures on all locally compact tions similar to those introduced above for ring formulas. fields have an deep underlying unity. The volumes of sets can be expressed ‘∀y. (∃x. x2 = y) =⇒ (∃m. 2m = val(y)).’ geometrically in a way that is independent of the underlying field. Moreover, there are effective procedures to calculate the varieties Xi and is syntactically correct. But the coefficients ai,Ni that represent the outer motivic volume at level m. If ‘∀f. ∀x. ∀y.f(y, ac(y))’ the outer ring formula approximations φm converge at some finite level m to the DVR formula φ, then we obtain effective procedures to calculate the is not well-formed, because quantifiers are not allowed over higher-order motivic volume of the formula. relations f in a first-order language. Also, ‘∀x ξ. (0 ≤ x) ∨ (ac(x) = ξ)’ 4. Applications and Conclusions is not well-formed, because of a type error; the variable symbol x appears What good is motivic measure? Here are a few examples. once as an integer 0 ≤ x and again as variable in the valued field ac(x). 4.1. Invariants of ring formulas. The group S is generated by vari- A syntactically correct formula is called a DVR formula. The aim of mot eties VarQ. Many geometrical invariants of varieties (such as Euler charac- motivic measure is to compute the “volume” of a DVR formula in a universal teristics and Hodge polynomials) can be reformulated as invariants of the way; that is, in a way that does not depend on the underlying locally compact ring S . This gives a novel way to attach invariants to every ring formula field. mot φ: take a geometric invariant of [φ] ∈ Smot. In particular, ring formulas have 3.3. Assumptions on the ring. The various examples that we have men- 2It is impossible for the structure K both to be locally compact and to have a residue tioned are all structures for the DVR language: rings of formal power series field k of characteristic zero, as required by Condition 3.3. In these final paragraphs, k[[t]], fields of formal Laurent series k((t)). For each prime p,(Q, ac, valp) is we allow the residue field to have positive characteristic. The residue field of a locally a structure for the language, as well as its completion (Qp, ac, valp). compact field is always finite. 16 THOMAS C. HALES WHAT IS MOTIVIC MEASURE? 13 The proof of this theorem uses quantifier elimination results to eliminate We will temporarily restrict the set of examples to structures (K, k, ac, val) the quantifiers that bind variables ranging over the valued field. It uses that satisfy the following conditions. results of Presburger on quantifier elimination to eliminate the quantifiers • K is a valued field of characteristic zero, with valuation function that range over the additive group of integers. The quantifiers that bind val : K → Z ∪ {∞} and angular component functions ac : K → k. variables in the residue field remain as quantifiers in the ring formula φm. • The residue field k has characteristic zero. • K is henselian. (We review the definition below.) 3.6.3. Scaling Factors. How is the scaling factor chosen in Equation 10 for Euclidean outer measures? The scaling factor 1/2nm is the unique constant Examples that satisfy these conditions include the fields k((t)), where k has that has the property that if the set A is itself a union of properly aligned characteristic zero. The analogy that will guides us is that these fields stand cubes (of width m0), then the outer measure of A is independent of m for in the same relation to locally compact DVR fields, as pseudo-finite fields all m ≥ m0. do to finite fields. To find the scaling factor for DVR formulas, we work a simple example 3.4. Henselian field. There is only one plausible definition for a henselian in which the DVR formula is itself a union of cubes of width m0 (that is, its field: A field is henselian if the field satisfies Hensel’s lemma. set of solutions is stable under perturbation of the power series tails). Hensel’s lemma gives checkable conditions on a polynomial that insure that it has a root in a given neighborhood. Hensel’s lemma occupies same Example 3.7. Let φ(x1, . . . , xn) = T, a formula that is true for all values ground in the realm of DVR rings that the intermediate value theorem of the free variables xi. In this case the outer ring formula approximation occupies in the realm of real numbers. (The intermediate value theorem is exact. Substitute polynomials p(ui·, t) for each xi and expand in terms of also gives checkable conditions on a polynomial that insure that it has a real mn distinct free variables uij to get root in a given neighborhood.) φm(uij) = T Our experience with motivic counting measures has alerted us to the im- portance of quantifier elimination, that is, the process of replacing a formula for all input values uij. The number of solutions of φm over a finite field nm with quantifiers ∀, ∃ with an equivalent formula that does not contain quan- Fq is q . If we take the motivic counting measure of φm, we find that the tifiers. The simplest case of quantifier elimination is the determination of variety that counts the points of φm over any finite field is the affine space of dimension nm: when there exists a root of a polynomial. Without a criterion for the ex- nm nm istence of roots to polynomials, quantifier elimination would be impossible. #A (Fq) = q . −1 For the pseudo-finite fields, this is handled through the defining property The class of φm in K[L ] ⊗ Q is of pseudo-finite fields that “every absolutely irreducible variety has a root.” [Anm] = [A1]nm = Lnm. For real fields, quantifier elimination is based on the intermediate value theorem. For henselian fields, quantifier elimination is based on Hensel’s From this one example, we see that the scaling factor for DVR formulas lemma. must be 1/Lnm. Lemma 3.3. (Hensel’s lemma) For every monic polynomial f ∈ K[x], Definition 3.8. Let φ be a DVR formula. Let the outer measure of φ at whose coefficients have non-negative valuation, and for every x such that level m be given by val(f(x)) > 0 [φm] ∈ K[L−1] ⊗ Q = S . and Lnm mot val(f 0(x)) = 0, This formula is analogous to Formula 10 for the Euclidean outer measure there exists y ∈ K such that f(y) = 0 and val(y − x) > 0. at level m. The numerator counts the number of centers of cubes that contain a solution to the DVR formula. This is stated as a lemma, but we view it as a condition on the field K and its valuation. It can be proved that the fields k((t)) and Qp are Definition 3.9. Let the motivic measure (or motivic volume) of φ be given henselian by showing that under the hypotheses of Hensel’s lemma, Newton’s by approximations to the roots −nm lim [φm]L , m→∞ x0 = x 0 whenever that limit exists. (The limit must be taken in a completion of xn+1 = xn − f(xn)/f (xn) Smot.) converge to a root. 14 THOMAS C. HALES WHAT IS MOTIVIC MEASURE? 15 3.5. Quantifier elimination. factor to use. Once we make these decisions, the formula for outer measure will take precisely the same form as Equation 10. Theorem 3.4. (Pas [14]) Let K be a field satisfying the other conditions In the planar case, we gave a construction of area of polygons as taking enumerated in 3.3 with residue field k . Let φ be a DVR formula. Then values in a scissor group S . The outer approximation of any bounded there is another formula φ0 without quantifiers of the valued field sort such poly planar set A by squares gives a value in the scissor group of polygons. Here that too, if our outer approximation to a DVR formula is with a ring formula, ∀(x, ξ, m) ∈ Kn × km × (Z ∪ {∞})r. φK (x, ξ, m) = φ0 K (x, ξ, m). then the value of the outer measure of the DVR formula will be in a scissor Moreover, the formula φ0 can be chosen to be independent of the structure ring Smot. K. Given all our preliminaries, it almost goes without saying at this point that rather that the counting of cubes that takes place in the numerator of 3.6. Outer measure of a DVR formula. As a first step toward con- Equation 10 will be replaced with the motivic counting measure of a ring structing the measure of a DVR formula, we will define an outer measure of formula. a formula. To motivate this construction, it might be helpful first to describe an analogous construction in Euclidean space. 3.6.2. Cubes. What is a cube? Well, it is a product of equal width intervals. In DVR formulas, a cube centered at a of “width” m is again a product of 3.6.1. An outer measure in Euclidean space. Fix a positive integer m. Tile intervals: Euclidean space with cubes of width 1/2m whose vertices are centered at m n points a with coordinates ai ∈ Z/2 . {(x1, . . . , xn) ∈ K | val(xi − ai) ≥ m, for i = 1, . . . , n}. According to the Calculus 101 approach to volume, we can approximate the volume of a set by counting the number of cubes that it meets. Let A If K = k[[t]], then the interval around a formal power series a is the set n of all formal power series with the same leading terms. Shaking (wagging) be a bounded set in R . Let Cm(A) be the set of cubes in this tiling that meet A. In our naive approach to measure, let us define the outer measure the tails of the power series fills out the interval. In other words, we can of A at level m to be make precise the idea of covering a DVR formula with cubes by replacing each solution to the DVR formula with a bigger set where the tails of the #Cm(A) (10) , solutions are allowed to vary. 2mn Let us make this precise. We have truncation map that is the number of cubes divided by the scaling factor 2mn. (If doing so did not involve logical circularity, we would identify 1/2mn with the volume k[[t]] → k[[t]]/(tm) ' km of cube and the entire expression as the volume of the set Cm(A) of cubes.) P∞ i Pm−1 i 0 ait 7→ 0 ait 7→ (a0, . . . , am−1). In the opposite direction, given b ∈ Km, there is a polynomial with those coefficients mX−1 i p(b, t) = bit ∈ k[[t]] 0 Definition 3.5. Let φ be a DVR formula with free variables (x1, . . . , xn) and no free variables of other sorts. An outer ring formula φm approximation to φ (at level m) is a ring formula in nm free variables uij such that over Figure 2. Volumes of DVR formulas can be approximated every field k: in Calculus 101 fashion by counting centers of cubes that {u ∈ knm | φ (u)} = meet a given formula, scaled according to the size of the m {u ∈ knm | ∃a , . . . , a . φ(a , . . . , a ) ∧ val(a − p(u , t)) ≥ m}. cubes. 1 n 1 n i ij This set is the set of centers of cubes that contain a solution to φ. The outer motivic measure of a DVR formula will be formed in an entirely analogous way. Of course, we will need to decide what to use for cubes, how Theorem 3.6. Outer ring formula approximations exist for every DVR to count the number of cubes that “meet” a given formula, and what scaling formula φ at every level m.