Existence of Bernstein-sato Polynomials by Using the Analytic Gröbner Fan

In 1987, C. Sabbah proved the existence of Bernstein-Sato polynomials associated with several analytic functions. The purpose of this article is to give a more elementary and constructive proof of the result of C. Sabbah based on the notion of the analytic Gröbner fan of a D-module. This paper is a translation of [Bah]. Introduction and statement of the main results Fix two integers n ≥ 1 and p ≥ 1 and v ∈ N r {0}. Let x = (x1, . . . , xn) and s = (s1, . . . , sp) be two systems of variables. Consider f1, . . . , fp ∈ C{x} = C{x1, . . . , xn} and denote by Dn the ring of differential operators with coefficients in C{x}. For b(s) ∈ C[s] = C[s1, . . . , sp], consider the following identity: (⋆) b(s)f s ∈ Dn[s]f , where f s+v = f s1+v1 1 · · · f sp+vp p . A polynomial b(s) satisfying such an identity is called a Bernstein-Sato polynomial (associated with f = (f1, . . . , fp)). The set of these polynomials form an ideal called the Bernstein-Sato ideal and denoted by B(f). Let us mention that usually, v is taken as (1, . . . , 1) or (0, . . . , 0, 1, 0, . . . , 0) where 1 is in the j-th position, j ∈ {1, . . . , p}. Let us give some historical recalls. In the case where p = 1 and f is a polynomial, I.N. Bernstein [Ber72] showed that the ideal B(f) is not zero (in this case, in (⋆), Dn is replaced with the Weyl algebra An(C), i.e. the ring of differential operators with polynomial coefficients). Again for p = 1 but in the analytic case, the fact that B(f) is not zero was proved by J. E. Björk [Bjö73] with similar methods as that of [Ber72]. In the same case, let us cite M. Kashiwara [Kas76] who published another proof and showed moreover that the unitary generator of the Bernstein-Sato ideal has rational roots. Now, for p ≥ 2 the proof in the polynomial case is an easy generalization of that by I. N. Bernstein, which can be found in [Lic88]. In the analytic case, the proof of the existence of a non zero Bernstein-Sato polynomial was given by C. Sabbah ([Sab87a] and [Sab87b]). Let us cite the contribution of A. Gyoja [Gyo93] who showed moreover that B(f) contains a rational non zero element. The goal of the present paper is a development of the proof by C. Sabbah. More precisely, we can decompose the proof of C. Sabbah into two main steps: the first one uses arguments similar to that used by M. Kashiwara in the case p = 1, the second one essentially consists in a finiteness result that reduces the problem to the first step. The second step of the proof by C. Sabbah is based on an “adapted fan”. The existence of such a fan is done in ([Sab87a] theor. A.1.1). Unfortunately, there is a gap in the proof of theorem A.1.1. (see also the comments after th. S1 in the present paper). In this paper,