Local topological algebraicity of analytic function germs

Abstract

T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

Local topological algebraicity with algebraic coefficients of analytic sets or functions

“…The proof is similar to the proof of Theorem 1.2 [BPR17] and so we will refer several times to this paper for details. For convenience x n−1 will denote the vector of indeterminates (x 1 , .…”

“…Let f n denote the product of the g s . Let ∆ n,i denote the i-th generalized discriminant of f n seen as a polynomial in x n (see 4.2 [BPR17]). This is a polynomial depending on a n−1 .…”

“…It shows that such solutions are C-points of a family of algebraic solutions defined over Q (see Theorem 7). In the next parts we apply this statement to prove Theorem 2 and then Theorem 1, essentially by proving that the approach used in [BPR17] remains valid in our situation.…”

“…Remark 5. In general for a given analytic map germ g : (K n , 0) −→ (K m , 0) there is no germ of a homeomorphism h : (K n , 0) −→ (K n , 0) such that g • h is the germ of a polynomial map: just take g : (K, 0) −→ (K 2 , 0) given by g(x) = (x, e x ) (see Example 6.3 [BPR17]). In particular Theorem 1 cannot be extended to analytic map germs (K n , 0) −→ (K m , 0).…”

“…Using Artin-Mazur Theorem we can reduce the situation to the case where ( V , 0) is the germ of an algebraic set and g is a unit u times a polynomial function germ P . Then, in [BPR17] is constructed a Thom stratification of the deformation…”