# Seminar of Algebra

### Bernstein's Ideal, relative filtration and slopes

Speaker:
Philippe Maisonobe (Université de Nice Sophia-Antipolis)
Email:
phm@unice.fr
Location:
Departamento de Álgebra
Date:
Fri, 20 may 2016 12:30
Bio:

Let $f_i: X \rightarrow {\bf C}$ be, for $i$ integer between $1$ and $p$, analytic functions on an analytic complex variety $X$ Let $F$ be the product of $f_i$ and if $\phi: X \rightarrow {\bf C}$ is a $C ^ {\infty}$ function with compact support in $K$, we denote: $$I_{\phi} (s_1, \ldots, s_p) = \int_X \mid f_1 (x) \mid ^ {s_1} \ldots \mid f_p (x) \mid ^ {s_p} \phi (x) \; dx \wedge d \overline {x} \; .$$ As in the case $p = 1$, by using the resolution's theorem of H. Hironaka, we can show that $I_ {\phi} (s_1, \ldots, s_p)$ extends to a meromorphic function with poles in hyperplanes of ${\bf C}^p$. F. Loeser studied these integrals and calls slope of $(f_1, \ldots, f_p)$ the directions of their polar hyperplanes.

Let ${\cal D} _X$ be the ring of differential operators and ${\cal D}_X [s_1, \ldots, s_p] = {\bf C}_X [s_1, \ldots, s_p] \otimes _ {\bf C} {\cal D} _X$. Let $m$ be a section of a holonomic ${\cal D}_X$-Module and $x_0 \in X$. We denote ${\cal B} (m, x_0, f_1, \ldots, f_p)$ the ideal formed by the polynomials $b \in { \bf C} [s_1, \ldots, s_p]$ such that near $x_0$ : $$b (s_1, \ldots, s_p) m f_1^ {s_1} \ldots f_p^ {s_p} \in {\cal D} _X [s_1, \ldots, s_p] \, m f_1^{s_1 + 1} \ldots f_p^{ s_p + 1} \; .$$

These polynomials are called Bernstein's polynomials of $(m, f_1, \ldots, f_p)$ in the neighborhood of $x_0$. Following J. Bernstein, this polynomials allow to build an extension of the integrals $I _ {\phi} (s_1, \ldots, s_p)$. C. Sabbah shows the existence for every $x_0 \in X$ of a finite set ${\cal H}$ of linear form such that: $$\prod_ {H \in {\cal H} } \prod_ {i \in I_ {\cal H}} (H (s_1, \ldots, s_p) + \alpha _ {H , i}) \in { \cal B} (m, x_0, f_1, \ldots, f_p) \; ,$$ where $\alpha _ {H,i}$ are complex numbers. The purpose of this talk is to prove the existence of a set ${\cal H}$ minimal.

In the first part of the talk, we will recall some classical results for $p=1$ and mainly the link between Bernstein's polynomials and vanishing cycles.