Frontier and Closure of a Semi-Pfa an Set

For a semi-Pfa an set, i.e., a real semianalytic set de ned by equations and inequalities between Pfa an functions in an open domain G, the frontier and closure in G are represented as semi-Pfa an sets. The complexity of this representation is estimated in terms of the complexity of the original set. Introduction. Pfa an functions introduced by Khovanskii [K] are analytic functions satisfying a triangular system of Pfa an di erential equations with polynomial coe cients (see De nition 1.1 below). The class of Pfa an 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 Pfa an. Khovanskii [K] found an e ective estimate for the number of isolated real solutions of a system of equations with Pfa an functions. This implies global niteness properties of semi-Pfa an sets, i.e., semianalytic sets de ned by equations and inequalities between Pfa an functions. Gabrielov [G1] found a similar estimate for the multiplicity of any complex solution of a system of Pfa an equations. The latter estimate allowed new niteness results for the geometry of semi-Pfa an sets in the real domain, including an e ective estimate on the exponent in the Lojasiewicz inequality for semi-Pfa an functions [G1] and the complexity of a weak strati cation of a semi-Pfa an set [GV], to be derived. The theory of Pfa an 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 de ned, outside coordinate hyperplanes, as Pfa an functions of the complexity depending only on the number of nonzero monomials, not on their degrees. This allows the techniques of Pfa an functions to be applied to the complexity of di erent operations with semialgebraic sets de ned by fewnomial expressions in terms of the number of nonzero monomials. Frontier and Closure of a Semi-Pfa an Set Page 2 In this paper we apply the niteness properties established in [G1] to construct a semi-Pfa an representation for the frontier and closure of a semi-Pfa an set. Note that the frontier and closure are considered only within the open domain where the Pfa an functions are de ned. We use a modi cation of the algorithm suggested in [G2] for the frontier and closure of a semianalytic set. For semi-Pfa an sets, this allows an e ective estimate of the complexity of the semi-Pfa an representation of the frontier and closure, in terms of the complexity of the original semi-Pfa an set. Using the estimates for the multiplicity of a Pfa an intersection from [G1], we reduce the question whether a given point x belongs to the closure of a semi-Pfa an set to the question whether x belongs to the closure of an auxiliary semialgebraic set Zx, replacing Pfa an functions by their nite-order Taylor expansions at x. We apply algebraic quanti er elimination [R, BPR] to obtain a semialgebraic condition on the coe cients of the polynomials in the formula de ning Zx, satis ed exactly when x belongs to the closure of Zx. As these coe cients are polynomial in x and in the values at x of the original Pfafan functions and their partial derivatives, the set of those x for which our semialgebraic condition is satis ed is semi-Pfa an. The paper is organized as follows. Section 1 introduces Pfa an functions and semiPfa an sets. The main result (Theorem 1.1) is formulated at the end of this section. Section 2 presents the necessary niteness properties of semi-Pfa an sets, based on the estimate of the multiplicity of Pfa an intersections from [G1]. Reduction to semialgebraic quanti er elimination and the proof of the main result are given in Section 3. Section 4 contains applications to fewnomial semialgebraic sets. 1. Pfa an functions and semi-Pfa an sets. Pfa an functions can be de ned 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 coe cients whenever the real or complex domain is considered. De nition 1.1. A Pfa an chain of order r 0 and degree 1 in an open domain G K is a sequence of analytic functions y1(x); : : : ; yr(x) in G satisfying a triangular system of Pfa an equations dyi(x) = n X j=1 Pij x; y1(x); : : : ; yi(x) dxj ; for i = 1; : : : ; r: (1) Here x = (x1; : : : ; xn), and Pij(x; y1; : : : ; yi) are polynomials of degree not exceeding . A function q(x) = Q(x; y1(x); : : : ; yr(x)), where Q(x; y1; : : : ; yr) is a polynomial of degree Frontier and Closure of a Semi-Pfa an Set Page 3 not exceeding > 0, is called a Pfa an function of degree with the Pfa an chain y1(x); : : : ; yr(x). Note that the Pfa an function q(x) is de ned only in the open domain G where all the functions yi(x) are analytic, even if q(x) itself can be extended as an analytic function to a larger domain. Remark. The above de nition of a Pfa an function corresponds to the de nition of a special Pfa an chain in [G1]. It is more restrictive than the de nitions in [K] and [G1] where Pfa an chains were de ned as sequences of nested integral manifolds of polynomial 1-forms. Both de nitions lead to (locally) the same class of Pfa an functions. In the following, we x a Pfa an chain y1(x); : : : ; yr(x) of degree de ned in an open domain G 2 K, and consider only Pfa an functions with this particular Pfa an chain, without explicit reference to the Pfa an chain, parameters n; r and , and the domain of de nition G. For K = R, we can de ne a complexi cation of the Pfa an chain y1(x); : : : ; yr(x), extending yi(x) as complex-analytic functions into an open domain C n containing G. Correspondingly, any Pfa an function can be extended to a complex Pfa an function in . We need the following simple properties of Pfa an functions [K, GV]. Lemma 1.1. The sum (resp. product) of two Pfa an functions, p1 and p2 of degrees 1 and 2 is a Pfa an function of degree max( 1; 2) (resp. 1 + 2). Lemma 1.2. A partial derivative of a Pfa an function of degree is a Pfa an function of degree + 1. Lemma 1.3. For a Pfa an function q(x) = Q(x; y1(x); : : : ; yr(x)) of degree , its Taylor expansion q (x; z) = X k:jkj 1 k1! kn! @q @xk (z)(x z) of order at z 2 G is a polynomial in x; z; y1(z); : : : ; yr(z) of degree not exceeding + . Proof. From Lemma 1.2, the value at z of a partial derivative @q=@x is a polynomial in z and y1(z); : : : ; yr(z) of degree at most + jkj( 1). Hence q (x; z) is a polynomial in x; z; y1(z); : : : ; yr(z) of degree at most + . De nition 1.2. An elementary semi-Pfa an set in G R of the format (L; n; r; ; ) is de ned by a system of equations and inequalities x 2 G; sign(qi(x)) = i; for i = 1; : : : ; L (2) Frontier and Closure of a Semi-Pfa an Set Page 4 where i 2 f 1; 0; 1g and qi(x) are Pfa an functions of degree not exceeding . A semi-Pfa an set of the format (N;L; n; r; ; ) is a nite union of at most N elementary semi-Pfa an sets of the format (L; n; r; ; ). Thus, a semi-Pfa an set of the format (N;L; n; r; ; ) can be de ned as