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Sturmfels, Bernd 击solving Systems of Polynomial Equations Citations From References: 47 Previous Up Next Article From Reviews: 4 MR1925796 (2003i:13037) 13P10 (14P99 14Q15 62-09 65H10) Sturmfels, Bernd FSolving systems of polynomial equations. (English summary) CBMS Regional Conference Series in Mathematics, 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. viii+152 pp. $32.00. ISBN 0-8218-3251-4 The book is about the aspects of systems of polynomial equations that are related to computational commutative algebra, discrete geometry, elimination theory, real geometry, partial differential equations, economics, probability, and statistics. The nice, lively exposition (reflecting a lecture course origin) gives a large number of snapshots of the vast panorama of interrelations between the listed disciplines. The text is full of actual implementations, documented as Maple, Matlab, Singular, Macaulay 2, PHCpack, and SOStools sessions. Virtually every page of the book shows the infectious enthusiasm of the author for the mathematics presented and the reader is exposed to the very frontiers of the current research in these areas. There are a number of exercises, problems, and conjectures. In short, the book is an excellent source for running seminars on any of the ten topics that correspond bijectively to its ten chapters. Chapter 1: Polynomials in one variable. The book starts with the basic concepts and facts on univariate polynomials—the fundamental theorem of algebra, discriminants, eigenvalues of the companion matrices (representing the multiplication by X on the quotient ring Q[X]/f(X)), Sturm’s theorem, Descartes’´ rule of signs, Puiseux series C{{t}}—setting the framework to be further developed in the next chapters for multivariate polynomials. Even in dimension one the technique of polyhedral geometry, the main vehicle of the computational algebraic geometry, manifests its relevance in computing the first term of the Puiseux series solutions. Hypergeometric series representations of roots of polynomials are introduced through Karl Mayer’s 1937 theorem, interpreting the roots as functions of the coefficients in terms of a certain system of second order linear partial differential equations. Explicit series are presented for a general quintic. Chapter 2: Grobner¨ bases of zero-dimensional ideals. This chapter concerns the case of poly- nomial systems representing zero-dimensional schemes and uses the technique of Grobner¨ bases. The initial observation is that standard monomials are exactly as many as (complex) zeros, count- ing multiplicities. The saturation operation enables one to enlarge (algorithmically) the given ideal in such a way that exactly one prespecified zero will disappear: to ease description of the zero-set one may need to get rid of certain zeros of high multiplicity. The same technique pro- vides a possibility to form a local ring at arbitrary zero and, thus, compute (algorithmically) its multiplicity. Higher analogues of the companion matrices, considered in Chapter 1, carry vital in- formation on the system of equations. These matrices admit a diagonalization by a simultaneous conjugation if and only if the ideal is radical. Yet another tool to attack a zero-dimensional poly- nomial ideal I is the trace form (with respect to a given Grobner¨ basis of I) of the bilinear pairing Q[X1,...,Xn]/I × Q[X1,...,Xn]/I → Q, defined by the products of two polynomials. It is a quadratic form whose signature is the number of real zeros (without counting multiplicities). Polynomial systems without solution are illustrated on a nice theorem of J. J. Duistermaat and W. van der Kallen [Indag. Math. (N.S.) 9 (1998), no. 2, 221–231; MR1691479 (2000k:22013)]: There is no Laurent multivariate complex polynomial whose powers all have 0 constant terms. In fact, the condition amounts to saying that the algebraic expressions of the constant coefficients in these multiples define a system without solution. Sturmfels conjectures an effective version of this theorem (in the one-dimensional case). Chapter 3: Bernstein’s theorem and Fewnomials. The material of this chapter has won for Sturmfels the 1999 Lester Ford prize for its beautiful exposition in [Amer. Math. Monthly 105 (1998), no. 10, 907–922; MR1656923 (99k:12003)]. Once again, here we have an attractive overview of the subject. By the classical Bezout´ Theorem the number of common zeros of two two-variable complex polynomials is at most the product of the degrees of these polynomials, provided the number is finite. A higher-dimensional analogue for the number of zeros in the complex torus (C∗)n for a generic system of complex polynomials with a prespecified set of support monomials is given by D. N. Bernstein’s theorem [Funkcional. Anal. i Prilozen.ˇ 9 (1975), no. 3, 1–4; MR0435072 (55 #8034)]. This result was in the early 1970s a kind of rebirth of the theory of Newton polytopes in the context of solving systems of polynomial equations. The theorem says that the number of zeros is bounded by the mixed volume of the sum of the Newton polytopes of the polynomials involved. (For two-variable polynomials such a bound is generically better than the product of degrees.) The main point is to relate, using “algebraic homotopy”, the given polynomial system to certain toric degeneration—a system whose equations are all given by binomials. For binomial systems the claim is easy to verify and, simultaneously, it is possible to control the number of solution branches during the homotopy (using Puiseux series). As an illustration, Sturmfels works out in detail the case of two variables, emphasizing the algorithmic nature of the approach. One can, in principle, approximate the solutions in the torus (C∗)n. The situation changes when we turn to the real case. A. G. Khovanski˘ı’s well-known theorem [Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 804–807; MR0600749 (82a:14006)] says that the number n (m) m of isolated real zeros in (R+) of n equations in n variables is bounded by 2 2 · (n + 1) , m being the total number of monomials involved. Notice that degrees of the polynomials are allowed to be arbitrarily big. In the proof one encounters, once again, algebraic homotopy of systems of equations. The degenerate case (t = 0) is easy to analyze and during the homotopy from t = 0 to t = 1 there are bifurcation points when pairs of new roots are born. The whole proof works because, based on the induction on m − n, the number of such bifurcation points is explicitly bounded. Historically, the motivation for Khovanski˘ı’s work was a more optimal upper bound, conjectured by Kushnirenko. This conjecture has recently been disproved in [B. Haas, Beitrage¨ Algebra Geom. 43 (2002), no. 1, 1–8; MR1913765 (2003d:14076)]. Sturmfels himself lost $500 in the hunt for optimal bounds [J. C. Lagarias and T. J. Richardson, Math. Intelligencer 19 (1997), no. 3, 9–15; MR1475144 (98i:14054)]. Chapter 4: Resultants. A homogeneous linear system of n equations with n variables has a nontrivial solution if and only if the determinant of the system vanishes. For systems of polynomial equations of higher degree the role of determinants is played by resultants—a central object of elimination theory. Sturmfels first considers the case of one variable. In particular, classical Sylvester and Bezoutian´ formulas are given and then the higher-dimensional case is elaborated in some detail. As far as Bernstein’s theorem relates to Bezout’s´ theorem one can develop a version of the classical resultant, called a sparse resultant [P. Pedersen and B. Sturmfels, Math. Z. 214 (1993), no. 3, 377–396; MR1245200 (94m:14068); I. M. Gel0fand, M. M. Kapranov and A. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhauser¨ Boston, Boston, MA, 1994; MR1264417 (95e:14045)] that only takes care of support monomials. There are some elegant formulas available for sparse resultants in certain special cases (due to Canni- Emiris, Dickenstein-Emiris, Eisenbud-Schreyer, and the author himself). These include the Chow forms of projective toric varieties (e.g., four trilinear forms), etc. Chapter 5: Primary decomposition. This chapter starts with generalities on primary decompo- sitions, illustrated on ideals in polynomial rings. The existing software Macaulay 2 and, most notably, Singular allow one to really compute such primary decompositions. In some special cases primary decompositions are particularly well-behaved, like for binomial ideals [D. Eisenbud and B. Sturmfels, Duke Math. J. 84 (1996), no. 1, 1–45; MR1394747 (97d:13031)], or complete in- tersections (Macaulay’s unmixedness theorem). Next comes a discussion on adjacent 2 × 2-minor ideals in n × m variables. Here one learns how nicely the general theory works for matrices of small size (results of Diaconis-Eisenbud-Sturmfels, Hosten, Shapiro). The chapter ends with a dis- cussion on permanental ideals. The permanent of an n × n matrix is the sum of all n! diagonal products. At present a satisfactorily complete theory (due to Laubenbacher-Swanson) only exists for 2 × 2-subpermanents. Chapter 6: Polynomial systems in economics. This chapter is devoted to Nash equilibria. The seminal ideas of John Nash half a century ago about how to express algebraically non-cooperative games, strategies, payoffs, and equilibria have been much publicized outside the professional world of mathematicians. The core of Nash’s model of a game is a system of multilinear polynomials of a very special type. One is interested in the real solutions to this system living in the region bounded by the common Newton polytope of the polynomials involved. This Newton polytope is the product of simplices. One can use Bernstein’s theorem (from Chapter 3) to compute the expected number of complex zeros of this system. What is important—according to a theorem of R. D. McKelvey and A. McLennan [J. Econom. Theory 72 (1997), no. 2, 411–425; MR1433526 (97j:90086)] —is that the expected number is in fact attained by counting only isolated real zeros in (the interior of) the critical Newton polytope.
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