Seminar of Algebra

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.