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Déformations à nombre de Milnor constant: Quelques résultats sur les polynômes de Bernstein. (Deformations of constant Milnor numbers: Some results on Bernstein polynomials). (French) Zbl 0721.32013

Compos. Math. 77, No. 2, 131-163 (1991); correction 78, No. 2, 239 (1991).
The author studies the Bernstein polynomial associated with a \(\mu\)- constant deformation. Let \(F(x_ 1,...,x_ n,y)\) be a holomorphic function defined in a polydisc \(W=X\times Y\subset {\mathbb{C}}^ n\times {\mathbb{C}}\) and let \({\mathcal D}_{W/Y}\) be the sheaf of relative differential operators i.e., \({\mathcal D}_{W/Y}={\mathcal O}_ W<\partial /\partial x_ 1,...,\partial /\partial x_ n>\). The author investigates certain \({\mathcal D}_{W/Y}\)-modules associated with the Bernstein polynomial and proves in particular the following result.
Theorem. Suppose that the function F(x,0) in x has an isolated critical point at \(x=0\). Then the following conditions are equivalent.
(i) F admits a \(\mu\)-constant deformation.
(ii) There exists an operator \(H=H_ r+...+H_ 1s^{r-1}+s^ r\in {\mathcal D}_{W/Y}[s]\) which annihilates \(F^ s\), where \(\deg (H_ j)\leq j.\)
(iii) \({\mathcal D}_{W/Y,0}[s]F^ s\) is a \({\mathcal D}_{W/Y,0}\)-module of finite type.
(iv) There exists a non-zero polynomial \(\delta\) (s) satisfying \(\delta (s)F^ s\in {\mathcal D}_{W/Y,0}F^{s+1}\).
Reviewer: S.Tajima (Niigata)

MSC:

32S30 Deformations of complex singularities; vanishing cycles
32C38 Sheaves of differential operators and their modules, \(D\)-modules
32S05 Local complex singularities
32G10 Deformations of submanifolds and subspaces
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References:

[1] I.N. Bernstein , The analytic continuation of generalised functions with respect to a parameter . Functional Anal. Appl. 6 (1972) 26-40. · Zbl 0282.46038 · doi:10.1007/BF01077645
[2] J.E. Björk , Dimensions over Algebras of differential operators . Preprint. Non publié (1974). · Zbl 0313.16027
[3] J. Briançon , Families équisingulières d’hypersurfaces à singularité isolée. Thèse de Doctorat d’Etat . Université de Nice (1976).
[4] J. Briançon , M. Granger , Ph. Maisonobe and M. Miniconi , Algorithme de calcul du polynôme de Bernstein: Cas non dégénéré . Ann. Inst. Fourier. Fasc. 3. Tome 39 (1989). · Zbl 0675.32008 · doi:10.5802/aif.1177
[5] F. Geandier , Polynômes de Bernstein et déformations à \mu -constant . C.R. Acad. Sci. Paris, t. 309, Série I, p. 831-834, (1989). · Zbl 0696.32010
[6] F. Geandier , Polynômes de Bernstein et déformations à \mu -constant . Thèse de doctorat. Université de Nice. (16 Juin 1989). · Zbl 0696.32010
[7] M. Kashiwara , B-Functions and Holonomic Systems . Invent. Math. 38 (1976) 33-53. · Zbl 0354.35082 · doi:10.1007/BF01390168
[8] F. Lazzeri , A theorem on the monodromy of isolated singularities . Singularités à Cargèse. (Astérisque 7 et 8). (1973). · Zbl 0301.32011
[9] Lê-Dung-Tràng , Topologie des singularités des hypersurfaces complexes . Singularités à Cargèse. (Astérisque 7 et 8). (1973). · Zbl 0331.32009
[10] Lê-Dung-Tràng And C.P. Ramanujam , The invariance of Milnor’s number implies the invariance of the topological type . Amer. J. Math. vol. 98 (1976) 67-78. · Zbl 0351.32009 · doi:10.2307/2373614
[11] B. Malgrange , Le polynôme de Bernstein d’une singularité isolée . Lect. Notes Math. vol. 459. Springer-Verlag (1975) 98-119. · Zbl 0308.32007
[12] B. Malgrange , Polynômes de Bernstein-Sato et cohomologie évanescente . Astérisque 101-102 (1983) 233-267. · Zbl 0528.32007
[13] B. Malgrange and M. Lejeune , Séminaire opérateurs différentiels et pseudo-différentiels , Preprint. Université de Grenoble (1975-76).
[14] J.H.M. Steenbrink , Mixed Hodge structure on the vanishing cohomology . Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn. 525-563 (1977). · Zbl 0373.14007
[15] B. Teissier , Cycles évanescents, sections planes et conditions de Whitney §II . Singularités à Cargèse. (Astérisque 7 et 8). (1973). · Zbl 0295.14003
[16] G.N. Tjurina , Locally semi-universal flat deformations of isolated singularities of complex spaces . Math. USSR. Izvestijà. Vol. 3 (1970) 967-999. · Zbl 0209.11301 · doi:10.1070/IM1969v003n05ABEH000814
[17] A.N. Varchenko , Gauss-Manin connection of isolated singular point and Bernstein polynomial . Bull. Sc. Math. 2èmesérie.104 (1980) 205-223. · Zbl 0434.32008
[18] A.N. Varchenko , Asymptotic Hodge structure in the vanishing cohomology . Math. USSR. Izvestijà. Vol. 18. no. 3 (1982). · Zbl 0489.14003 · doi:10.1070/IM1982v018n03ABEH001395
[19] T. Yano , On the theory of b-functions . Publ. RIMS, Kyoto University, 14 (1978) 111-202. · Zbl 0389.32005 · doi:10.2977/prims/1195189282
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