Some Results on Bernstein-sato Polynomials for Parametric Analytic Functions

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. We also make an extensive study of an example for which we give an expression of a generic (and under some conditions, a relative) Bernstein-Sato polynomial. Let X ⊂ C and Y ⊂ C be compact polydiscs centered at the origin, Z = X × Y and f = (f1, . . . , fp) (p ≥ 2) an analytic map from X to C. We are interested in the study of Bernstein-Sato polynomials of f(x, y0) when y0 moves through Y . Our work is related to the notion of generic Bernstein-Sato polynomials as in Briançon et al. [10] (for p = 1) and Biosca [7]. Herein we shall adopt a more constructive method as in Bahloul [4] (where the case p = 1 was treated), based on the first part [5] and Bahloul [2]. Our goal is to give analogous results to [4]. However, since the construction in [2] is entirely algorithmic only when p = 2, a part of the results herein shall be shown only for p = 2. It would be a nice result if one could wholly achieve [2] in an algorithmic way (here “algorithmic” means “in an infinite way”). Note that a similar question was treated in the case of polynomials fj in Bahloul [1] with direct methods while constructive methods were used in Leykin [16] (for p = 1) and Briançon, Maisonobe [13] (for p ≥ 1). Note. If OCn+m denotes the sheaf of analytic functions on C, we shall identify OZ with the germ OCn+m,0. Sometimes, we will reduce Z without an explicit mention so that OZ shall be identified with the set OCn+m(U) of sections of OCn+m on an open (poly)disc 0 ∈ U ⊂ Z.