Complexity of Computations with Pfaffian and Noetherian Functions Andrei GABRIELOV Department of Mathematics Purdue University West Lafayette, IN 47907, USA Nicolai VOROBJOV Department of Computer Science University of Bath Bath, BA2 7AY, England, UK Abstract This paper is a survey of the upper bounds on the complexity of basic algebraic and geometric operations with Pfaffian and Noetherian functions, and with sets defin- able by these functions. Among other results, we consider bounds on Betti numbers of sub-Pfaffian sets, multiplicities of Pfaffian intersections, effectiveLojasiewicz inequality for Pfaffian functions, computing frontier and closure of restricted semi-Pfaffian sets, con- structing smooth stratifications and cylindrical cell decompositions (including an effective version of the complement theorem for restricted sub-Pfaffian sets), relative closures of non-restricted semi-Pfaffian sets and bounds on the number of their connected compo- nents, bounds on multiplicities of isolated solutions of systems of Noetherian equations. 1 Introduction Pfaffian functions, introduced by Khovanskii [24, 25] at the end of the 1970s, are analytic functions satisfying triangular systems of Pfaffian (first order partial differential) equations with polynomial coefficients. Over R, these functions, and the corresponding semi- and sub- Pfaffian sets, are characterized by global finiteness properties similar to the properties of polynomials and semialgebraic sets. This allows one to establish efficient upper bounds on the complexity of different algebraic and geometric operations with these functions and sets. One of important applications of the Pfaffian theory is in the real algebraic geometry of fewnomials — polynomials defined by simple formulas, possibly of a high degree. The complexity of operations with fewnomials in many cases allows upper bounds in terms of the complexity of the defining formulas, independent of their degree. Over C, Pfaffian functions, and more general Noetherian functions (satisfying the same kind of equations but without the triangularity condition) are characterized by local finiteness properties, sufficient for upper bounds on the complexity of stratification, frontier and closure. 1 2 A. Gabrielov, N. Vorobjov This paper is a survey of results about the upper bounds for operations on Pfaffian and Noetherian functions, and on sets definable by these functions. All bounds are functions of a finite set, called format, of some natural parameters (like the degree or the number of variables) associated with a Pfaffian or Noetherian function or a definable set. The goal is to obtain as low upper bounds as possible. The content of the paper is as follows. We start with definitions and examples of Pfaffian functions (including fewnomials), semi- and sub-Pfaffian sets. We discuss Khovanskii’s bound on the number of isolated solutions of a system of Pfaffian equations, and its extensions for Betti numbers of semi- and sub-Pfaffian sets. Next, we derive an upper bound for the multiplicity of a Pfaffian intersection. This bound is then used to obtain an effective version ofLojasiewicz inequality for a Pfaffian function, and an algorithm for constructing frontier and closure (in its domain of definition) of a semi- Pfaffian set. The bound for Pfaffian multiplicities is also used to construct an algorithm for a weak stratification of a semi-Pfaffian set, i.e., for a representation of the set as a disjoint union of smooth manifolds. The complexity (running time) of the stratification algorithm is explicitly estimated in terms of the format of the input semi-Pfaffian set. This also implies explicit upper bounds on the number of strata and their formats. Stratification is used for an effective proof of the following complement theorem: the complement of a projection of a restricted (relatively compact in its domain of definition) semi-Pfaffian set to a subspace is again a projection of a semi-Pfaffian set. The proof of the theorem uses an algorithm for a cylindrical cell decomposition of a restricted semi-Pfaffian set, a construction which is importantinitsownright. For general (not necessarily restricted) semi-Pfaffian sets, the complement theorem is not known to be true. We describe a wider category of sets, called limit sets, which is a Boolean algebra and is also closed under projections to subspaces. We describe an explicit upper bound on the number of connected components of a limit set in terms of the format of the set. Noetherian functions do not generally satisfy the global finiteness properties of Pfaffian functions. However, it is possible to establish some local finiteness properties. We describe in some detail a proof of an explicit upper bound on the multiplicity of an isolated solution of a system of Noetherian equations. This proof involves several stages. First, we give a brief introduction to integration over Euler characteristics. Next, we consider the univariate case, which implies, in particular, an upper bound on the vanishing order of a multivariate Noetherian function. As another application of the univariate result, we derive an upper bound for the degree of nonholonomy of a system of polynomial vector fields. The last stage of the proof for the multivariate case involves a lower bound on codimension of the set of intersections of high multiplicity. 2 Pfaffian functions and sub-Pfaffian sets Pfaffian functions, introduced by Khovanskii at the end of the 1970s, are real or complex analytic functions satisfying triangular systems of Pfaffian (first order partial differential) equations with polynomial coefficients. We use the notation Kn,whereK is either R or C, in the statements relevant to both cases. Complexity of Computations with Pfaffian and Noetherian Functions 3 2.1 Definition [24, 25, 18] A Pfaffian chain of the order r 0 and degree α 1inan n ≥ ≥ open domain G K is a sequence of analytic functions f1,... ,fr in G satisfying differential equations ⊂ df j(x)= gij(x,f1(x),... ,fj(x))dxi (2.1) 1 i n ≤X≤ for 1 j r.Heregij(x,y1,... ,yj)arepolynomialsinx =(x1,... ,xn), y1,... ,yj of ≤ ≤ degrees not exceeding α. A function f(x)=P (x,f1(x),... ,fr(x)), where P (x,y1,... ,yr) is a polynomial of a degree not exceeding β 1, is called a Pfaffian function of order r and degree (α, β). Note that the Pfaffian function≥ f is defined only in the domain G where all functions f1,... ,fr are analytic, even if f itself can be extended as an analytic function to a larger domain. 2.2 Remark This definition is more restrictive than the definition from [25], where the Pfaffian chains are defined as sequences of nested integral manifolds of polynomial 1-forms. Both definitions lead to essentially the same class of Pfaffian functions, although the orders and degrees of Pfaffian chains for the same Pfaffian function can be different according to these two definitions. We found Definition 2.1 to be more convenient to trace the behaviour of parameters of Pfaffian functions under different operations. More general definitions of Pfaffian functions, where the coefficients of (2.1) are not nec- essarily polynomial, were considered in [29, 32]. Most of our constructions can be adjusted to this more general definition, however upper bounds on the complexity may not be efficient enough in this case. 2.3 Example (a) Pfaffian functions of order 0 and degree (1,β) are polynomials of degrees not exceeding β. (b) The exponential function f(x)=eax is a Pfaffian function of order 1 and degree (1, 1) in R, due to the equation df (x)=af(x)dx. More generally, for i =1, 2,... ,r,let Ei 1(x) Ei(x):=e − , E0(x)=ax.ThenEr(x) is a Pfaffian function of order r and degree (r, 1), since dEr(x)=aE1(x) Er(x)dx. ··· (c) The function f(x)=1/x is a Pfaffian function of order 1 and degree (2, 1) in the domain x R x =0 , due to the equation df (x)= f 2(x)dx. { ∈ | 6 } − (d) The logarithmic function f(x)=ln(x ) is a Pfaffian function of order 2 and degree (2, 1) | | in the domain x R x =0 , due to equations df (x)=g(x)dx and dg(x)= g2(x)dx, where g(x)=1{/x.∈ | 6 } − (e) The polynomial f(x)=xm can be viewed as a Pfaffian function of order 2 and degree (2, 1) in the domain x R x =0 (but not in R), due to the equations df (x)= mf(x)g(x)dx and dg(x{)=∈ g|2(x)6 dx,where} g(x)=1/x. In some cases a better way to deal with xm is to change the− variable x = eu reducing this case to (b). (f) The function f(x)=tan(x) is a Pfaffian function of order 1 and degree (2, 1) in the domain x x = π/2+kπ , due to the equation df (x)=(1+f 2(x))dx. k Z R ∈ { ∈ | 6 } (g) The functionT f(x) = arctan(x) is a Pfaffian function in R of order 2 and degree (3, 1), due to equations df (x)=g(x)dx and dg(x)= 2xg2(x)dx,whereg(x)=(x2 +1) 1. − − 4 A. Gabrielov, N. Vorobjov (g) The function cos(x) is a Pfaffian function of order 2 and degree (2, 1) in the domain x x = π +2kπ , due to equations cos(x)=2f(x) 1, df (x)= f(x)g(x)dx, k Z R ∈ { ∈ | 1 6 2 } 2 − − and dg(x)= 2 (1 + g (x))dx,wheref(x)=cos (x/2) and g(x)=tan(x/2). Also, since cos(T x) is a polynomial of degree m of cos(x/m), the function cos(x)isPfaffianoforder 2 and degree (2,m) in the domain x x = mπ +2kmπ . The same is true, of k Z R course, for any shift of this domain by∈ a{ multiple∈ | of6 π.