Polynômes de Bernstein-Sato à plusieurs variables. (Bernstein-Sato polynomials in several complex variables). (French) Zbl 0634.32003

Sémin., Équations Dériv. Partielles 1986-1987, Exp. No. 19, 6 p. (1987).
This paper is a summary of two papers of the author [“Proximité évanescente. I, II.”, Compos. Math. 62, 283-328 (1987; Zbl 0622.32012), ibid. 64, 213-241 (1987; Zbl 0632.32006)]. The author first recalls the existence of the Bernstein-Sato polynomial for an analytic complex function or for a polynomial f. Given more than one function, \(f_ 1,...,f_ p\), and an holonomic distribution u he obtains similarly functional equations of the type: \[ B_ k(s_ 1,...,s_ p) f_ 1^{s_ 1}...f_ p^{s_ p}u = P_ k(x,\partial x,u) f_ 1^{s_ 1}...f_ k^{s_ k+1}f_ p^{s_ p}u. \] Here \(B_ k\) is a polynomial in s, product of affine factors depending on a finite set \({\mathcal L}\) of linear parts \(L(s_ 1,...,s_ k)\). This result allows him, for example, to prove that the integrals: \[ I(s,\sigma)=\int_{{\mathbb{C}}^ n}| f|^{2s} | g|^{2\sigma} dx\wedge dx \] can be extended as meromorphic functions with poles along lines with rational coefficients. When u generates a regular holonomic module, it can be proved \((2^{nd}\) paper quoted above) that the set \({\mathcal L}\) of linear forms depends only on the characteristic variety of \({\mathcal D}u\). The proof of the first result is sketched at the end of the paper. It relies on the introduction of V- filtrations indexed by \({\mathbb{Z}}^ p\) of \({\mathcal D}\)-modules with support on the graph of \((f_ 1,...,f_ p)\), and on the Rees’ rings and modules associated to these filtrations. The key point of this proof uses a theorem similar to the “flatenning theorem” of Hironaka proved by the author and F. Castro in the same papers.
Reviewer: J.M.Granger


32A20 Meromorphic functions of several complex variables
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32C30 Integration on analytic sets and spaces, currents
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