Frontier and Closure of a Semi-Pfaffian Set

For a semi-Pfaffian set, i.e., a real semianalytic set defined by equations and inequalities between Pfaffian functions in an open domain G, the frontier and closure in G are represented as semi-Pfaffian sets. The complexity of this representation is estimated in terms of the complexity of the original set. Introduction. Pfaffian functions introduced by Khovanskii [K] are analytic functions satisfying a triangular system of Pfaffian differential equations with polynomial coefficients (see Definition 1.1 below). The class of Pfaffian functions includes elementary functions, such as the exponential and logarithmic function, and trigonometric functions in a bounded domain. Many important special functions, such as elliptic integrals, are also Pfaffian. Khovanskii [K] found an effective estimate for the number of isolated real solutions of a system of equations with Pfaffian functions. This implies global finiteness properties of semi-Pfaffian sets, i.e., semianalytic sets defined by equations and inequalities between Pfaffian functions. Gabrielov [G1] found a similar estimate for the multiplicity of any complex solution of a system of Pfaffian equations. The latter estimate allowed new finiteness results for the geometry of semi-Pfaffian sets in the real domain, including an effective estimate on the exponent in the Lojasiewicz inequality for semi-Pfaffian functions [G1] and the complexity of a weak stratification of a semi-Pfaffian set [GV], to be derived. The theory of Pfaffian functions has an important application to computations with usual polynomial functions, based on Khovanskii’s theory of “fewnomials.” Fewnomials are polynomials containing a limited number of nonzero monomials, of an arbitrarily high degree. Fewnomials can be defined, outside coordinate hyperplanes, as Pfaffian functions of the complexity depending only on the number of nonzero monomials, not on their degrees. This allows the techniques of Pfaffian functions to be applied to the complexity of different operations with semialgebraic sets defined by fewnomial expressions in terms of the number of nonzero monomials. Frontier and Closure of a Semi-Pfaffian Set Page 2 In this paper we apply the finiteness properties established in [G1] to construct a semi-Pfaffian representation for the frontier and closure of a semi-Pfaffian set. Note that the frontier and closure are considered only within the open domain where the Pfaffian functions are defined. We use a modification of the algorithm suggested in [G2] for the frontier and closure of a semianalytic set. For semi-Pfaffian sets, this allows an effective estimate of the complexity of the semi-Pfaffian representation of the frontier and closure, in terms of the complexity of the original semi-Pfaffian set. Using the estimates for the multiplicity of a Pfaffian intersection from [G1], we reduce the question whether a given point x belongs to the closure of a semi-Pfaffian set to the question whether x belongs to the closure of an auxiliary semialgebraic set Zx, replacing Pfaffian functions by their finite-order Taylor expansions at x. We apply algebraic quantifier elimination [R, BPR] to obtain a semialgebraic condition on the coefficients of the polynomials in the formula defining Zx, satisfied exactly when x belongs to the closure of Zx. As these coefficients are polynomial in x and in the values at x of the original Pfaffian functions and their partial derivatives, the set of those x for which our semialgebraic condition is satisfied is semi-Pfaffian. The paper is organized as follows. Section 1 introduces Pfaffian functions and semiPfaffian sets. The main result (Theorem 1.1) is formulated at the end of this section. Section 2 presents the necessary finiteness properties of semi-Pfaffian sets, based on the estimate of the multiplicity of Pfaffian intersections from [G1]. Reduction to semialgebraic quantifier elimination and the proof of the main result are given in Section 3. Section 4 contains applications to fewnomial semialgebraic sets. 1. Pfaffian functions and semi-Pfaffian sets. Pfaffian functions can be defined in the real or complex domain. We use the notation K, where K is either R or C, in the statements relevant to both cases. Correspondingly, “analytic” means real or complex analytic, and “polynomial” means a polynomial with real or complex coefficients whenever the real or complex domain is considered. Definition 1.1. A Pfaffian chain of order r ≥ 0 and degree α ≥ 1 in an open domain G ⊂ K is a sequence of analytic functions y1(x), . . . , yr(x) in G satisfying a triangular system of Pfaffian equations