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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2002, Volume 238, Pages 86–96
(Mi tm346)
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This article is cited in 3 scientific papers (total in 3 papers)
Équations fonctionnelles associées à des fonctions analytiques
J. Briançon, Ph. Maisonobe, M. Merlea a Université de Nice Sophia Antipolis
Abstract:
Let XX be a XX, and denote by F=f1…fpF=f1…fp their product. Given a regular holonomic DXDX-module MM and a section m∈Mm∈M, denote by B(x,f1,…,fp,m)B(x,f1,…,fp,m) the Bernstein–Sato ideal of C[s1,…,sp]C[s1,…,sp] consisting of polynomials b(s1,…,sp)b(s1,…,sp) such that there exists, in a neighborhood of x∈F−1(0)x∈F−1(0), a differential operator P(s1,…,sp)∈DX⊗CC[s1,…,sp]P(s1,…,sp)∈DX⊗CC[s1,…,sp] satisfying P(s1,…,sp)mfs1+11…fsp+1p=b(s1,…,sp)mfs11…fsppP(s1,…,sp)mfs1+11…fsp+1p=b(s1,…,sp)mfs11…fspp. Claude Sabbah proved that this ideal is nonzero. One can associate to the characteristic variety of the DX[s1,…,sp]-module DX[s1,…,sp]mfs11…fspp a finite set Hf,m of hyperplanes in Cp. We prove that there exists a Bernstein–Sato polynomial (i.e., a nonzero member of the Bernstein–Sato ideal) which is a product of one variable polynomials if and only if the set Hf,m is contained in the union of the coordinate hyperplanes. In the two variables case (p=2) we prove that there exist a Bernstein–Sato polynomial the higher degree form of which vanishes on and only on the set Hf,m.
Received in November 2000
Citation:
J. Briançon, Ph. Maisonobe, M. Merle, “Équations fonctionnelles associées à des fonctions analytiques”, Monodromy in problems of algebraic geometry and differential equations, Collected papers, Trudy Mat. Inst. Steklova, 238, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 86–96; Proc. Steklov Inst. Math., 238 (2002), 77–87
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https://www.mathnet.ru/eng/tm346 https://www.mathnet.ru/eng/tm/v238/p86
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