Estimates for zero loci of Bernstein-Sato ideals (Q6619104)

Let \(F=(f_1,\cdots, f_r)\) be a family of polynomials in \(n\) variables with complex coefficients, with \(r>0\). To each \(a=(a_1,\cdots, a_r)\in\mathbb{N}^r\) such that \(f_1^{a_1}\cdots f_r^{a_r}\) admits zeros in \(\mathbb{C}^n\) one associates the Bernstein-Sato ideal \(B^a_F\) of \(\mathbb{C}[s_1,\cdots, s_r]\). The purpose of this paper is twofold:\N\NIn Theorem 1.2 the authors prove a refinement of the already known characterization of the 1-codimensional irreducible components of the zero locus of \(B^a_F\) (\(Z(B_F^a)\)) by means of log resolutions. Such a refinement implies upper bounds for \(Z(B_F^a)\).\N\NIn the second part (Section 4, with main result Theorem 1.4), using the language of polytopes, the authors obtain lower bounds for the same zero locus.\N\NThese results have an analytic version by considering germs of analytic morphisms.