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Zbl 0721.32013
Geandier, Françoise
Déformations à nombre de Milnor constant: Quelques résultats sur les polynômes de Bernstein. (Deformations of constant Milnor numbers: Some results on Bernstein polynomials). (French)
[J] Compos. Math. 77, No.2, 131-163; correction 78, No.2, 239 (1991). ISSN 0010-437X

The author studies the Bernstein polynomial associated with a $\mu$- constant deformation. Let $F(x\sb 1,...,x\sb n,y)$ be a holomorphic function defined in a polydisc $W=X\times Y\subset {\bbfC}\sp n\times {\bbfC}$ and let ${\cal D}\sb{W/Y}$ be the sheaf of relative differential operators i.e., ${\cal D}\sb{W/Y}={\cal O}\sb W<\partial /\partial x\sb 1,...,\partial /\partial x\sb n>$. The author investigates certain ${\cal D}\sb{W/Y}$-modules associated with the Bernstein polynomial and proves in particular the following result. \par Theorem. Suppose that the function F(x,0) in x has an isolated critical point at $x=0$. Then the following conditions are equivalent.\par (i) F admits a $\mu$-constant deformation. \par (ii) There exists an operator $H=H\sb r+...+H\sb 1s\sp{r-1}+s\sp r\in {\cal D}\sb{W/Y}[s]$ which annihilates $F\sp s$, where $\deg (H\sb j)\le j.$ \par (iii) ${\cal D}\sb{W/Y,0}[s]F\sp s$ is a ${\cal D}\sb{W/Y,0}$-module of finite type. \par (iv) There exists a non-zero polynomial $\delta$ (s) satisfying $\delta (s)F\sp s\in {\cal D}\sb{W/Y,0}F\sp{s+1}$.
[S.Tajima (Niigata)]

MSC 2000:: *32S30 Deformations of singularities (analytic spaces)
32C38 Sheaves of differential operators (analytic spaces)
32S05 Local singularities (analytic spaces)
32G10 Deformations of submanifolds

Keywords: ${\cal D}$-module; Bernstein polynomial; $\mu $ -constant deformation; sheaf of relative differential operators; isolated critical point

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