过滤相对论，伯恩斯坦等人的理想
Filtration Relative, l'Idéal de Bernstein et ses pentes
摘要：
设$ f_i: X \rightarrow {\bf C}$是定义在复解析簇$X$上的解析函数，其中$i$是$ 1$和$ p $之间的整数.让我们考虑$ {\cal D}_X $的线性微分算子和$ {\cal D}_X [s_1, \ldots, s_p] = {\bf C}_X [s_1, \ldots, s_p] \otimes_ {\bf C} {\cal D}_X$的环。设$ m $是完整$ {\cal D}_X $ -模的一部分。我们将$ {\cal B}(m, x_0, f_1, \ldots, f_p) $表示为$ { \bf C} [s_1, \ldots, s_p] $的理想，该理想由在$ x_0 \in X$的邻域内满足以下条件的多项式$ b $构成：$$ B (s_1, \ldots, s_p) m f_1^{ s_1} \ldots f_p ^{s_p} \in {\cal D}_X [s_1, \ldots, s_p] \, m f_1^{s_1 + 1} \ldots f_p^{s_p + 1} \; . $$这个理想被称为伯恩斯坦理想。C. Sabbah表明存在的每一个$ x_0 \in X $的有限集$ {\cal H} $的线性形式的系数在$ {\bf N} $，这样：$$ \prod_{H \in {\cal H}} \prod_{i \in I_{\cal H}} (H (s_1, \ldots, s_p) + α_{H , i}) \in {\cal B} (m, x_0, f_1, \ldots, f_p) \; , $$其中$α_{H,i} $是复数。这篇文章的目的是显示特别是存在一个最小集$ {\cal H} $。此外，当$ m $是一个完整的定期${\cal D}_X$ -模块的一部分，我们将精确几何这一套从特征品种的${\cal D}_X$ -模块产生的$m$。我们特别介绍和研究了与我们问题有关的$ {\cal D}_X [s_1, \ldots, s_p] $ -模的相关特征簇。这允许指定伯恩斯坦理想的结构。
英文摘要：
Let $ f_i: X \rightarrow {\bf C}$, for $i$ integer between $ 1$ and $ p $, be analytic functions defined on a complex analytic variety $X$. Let us consider $ {\cal D}_X $ the ring of linear 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. We denote $ {\cal B}(m, x_0, f_1, \ldots, f_p) $ the ideal of $ { \bf C} [s_1, \ldots, s_p] $ constituted by the polynomials $ b $ satisfying in the neighborhood of $ x_0 \in X$ : $$ B (s_1, \ldots, s_p) m f_1^{ s_1} \ldots f_p ^{s_p} \in {\cal D}_X [s_1, \ldots, s_p] \, m f_1^{s_1 + 1} \ldots f_p^{s_p + 1} \; . $$ This ideal is called Bernstein's ideal. C. Sabbah shows the existence for every $ x_0 \in X $ of a finite set $ {\cal H} $ of linear forms with coefficients in $ {\bf N} $, such that: $$ \prod_{H \in {\cal H}} \prod_{i \in I_{\cal H}} (H (s_1, \ldots, s_p) + α_{H , i}) \in {\cal B} (m, x_0, f_1, \ldots, f_p) \; , $$ where $α_{H,i} $ are complex numbers. The purpose of this article is to show in particular the existence of a minimal set $ {\cal H} $. In addition, when $ m $ is a section of a holonomic regular ${\cal D}_X$-Module, we will precise geometrically this set from the characteristic variety of ${\cal D}_X$-Module generated by $m$. We introduce and study especially the relative characteristic variety of the $ {\cal D}_X [s_1, \ldots, s_p] $ - Modules related to our problem. This allows to specify the structure of the Bernstein's ideals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
