\'Equations fonctionnelles associ\'ees \`a des fonctions analytiques

J. Briançon ; Ph. Maisonobe ; M. Merle

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 86-96.

Résumé

Let $X$ be a  $X$, and denote by $F=f_1\dots f_p$ their product. Given a regular holonomic $\mathcal D_X$-module $\mathcal M$ and a section $m\in \mathcal M$, denote by $B(x,f_1,\dots ,f_p,m)$ the Bernstein–Sato ideal of $\mathbf C[s_1,\dots, s_p]$ consisting of polynomials $b(s_1,\dots ,s_p)$ such that there exists, in a neighborhood of $x\in F^{-1}(0)$, a differential operator $P(s_1,\dots,s_p)\in \mathcal D_X \otimes _{\mathbf C}\mathbf C[s_1,\dots , s_p]$ satisfying $P(s_1,\dots ,s_p)m f_1^{s_1+1}\dots f_p^{s_p+1} =b(s_1,\dots ,s_p)m f_1^{s_1}\dots f_p^{s_p}$. Claude Sabbah proved that this ideal is nonzero. One can associate to the characteristic variety of the $\mathcal D_X[s_1,\ldots ,s_p]$-module $\mathcal D_X[s_1,\ldots,s_p]m f_1^{s_1}\dots f_p^{s_p}$ a finite set ${\mathcal H}_{f,m}$ of hyperplanes in $\mathbf C^p$. 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 $\mathcal H_{f,m}$ 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 $\mathcal H_{f,m}$.

Bibliographie

[1] Biosca H., “Caractérisation de l'existence de polynômes de Bernstein relatifs associés à une famille d'applications analytiques”, C. R. Acad. Sci. Paris. Sér. 1, 325:4 (1997), 395–398 | MR | Zbl

[2] Björk J.-E., Rings of differential operators, North-Holland, Amsterdam, 1979 | MR | Zbl

[3] Briançon J., Laurent Y., Maisonobe Ph., “Sur les modules différentiels réguliers cohérents relativement à une projection”, C. R. Acad. Sci. Paris. Sér. 1, 313:5 (1991), 285–288 | MR | Zbl

[4] Briançon J., Maisonobe Ph., Merle M., “Constructibilité de l'idéal de Bernstein”, Combinatorial methods in representation theory, Adv. Stud. Pure Math., 28, Kinokuniya, Tokyo, 2000, 79–95

[5] Briançon J., Maisonobe Ph., Merle M., “Éventails associés à des fonctions analytiques”, Tr. Mat. Inst. Steklova, 70–80 | MR

[6] Biosca H., Briançon J., Maisonobe Ph., Maynadier H., “Espaces conormaux relatifs. II: Modules différentiels”, Publ. Res. Inst. Math. Sci., 34 (1998), 123–134 | DOI | MR | Zbl

[7] Briançon J., Granger M., Maisonobe Ph., “Sur les systèmes différentiels relativement spécialisables et l'existence d'équations fonctionnelles relatives”, Bull. Soc. Math. France, 124:2 (1996), 217–242 | MR | Zbl

[8] Granger M., Maisonobe Ph., “A basic course on differential modules”, $\mathcal D$-modules cohérents et holonomes: Cours du CIMPA, Travaux en cours, 45, Hermann, Paris, 1993, 103–168 | MR | Zbl

[9] Kashiwara M., Kawai T., “On holonomic systems for $\prod_{l=1}^{N}(f_l+\sqrt{-1}0)^{\lambda_l}$”, Publ. RIMS. Kyoto Univ., 15 (1979), 551–575 | DOI | MR | Zbl

[10] Sabbah C., “Proximité évanescente. I: La structure polaire d'un $\mathcal D$ module”, Compos. Math., 62 (1987), 283–328 | MR | Zbl

[11] Sabbah C., “Proximité évanescente. II: Équations fonctionnelles pour plusieurs fonctions analytiques”, Compos. Math., 64 (1987), 213–241 | MR | Zbl

[12] Sabbah C., Appendice à Proximité évanescente, II, Centre Math. École Polytech, Palaiseau, 1988

[13] Schapira P., Microdifferential systems in the complex domain, Grundl. Math. Wissensch., 269, Springer, Berlin etc., 1985 | MR | Zbl