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Fonctions \(| f| ^ S\), théorie de Hodge et polynômes de Bernstein-Sato. \((| f| ^ s\) functions, Hodge theory and Bernstein-Sato polynomials). (French) Zbl 0621.32018
Géométrie algébrique et applications, C. R. 2ième Conf. int., La Rabida/Espagne 1984, III: Géométrie réelle. Systèmes différentiels et théorie de Hodge, Trav. Cours 24, 21-33 (1987).
[For the entire collection see Zbl 0614.00007.]
From author’s summary: ”In this article, an analogy, not well understood up to now, is shown between the asymptotic estimates of the number of approximate p-adic solutions for a diophantian equation and differential equations associated to it.
In the first part, fundamental results of J. I. Igusa on complex powers \(| f|^ s\) and some more recent results are reviewed. A new estimate for the first pole of \(| f|^ s\) is given in terms of polar invariants; the proof uses transcendental methods (Hodge theory).
In the second part, the complex case is described, especially the relation with Bernstein-Sato polynomial and volume of tubes. Hodge theory is also crucial here, mainly with the works of Schmid, Steenbrink and Varchenko.”
Reviewer: D.Barlet

MSC:
32S05 Local complex singularities
32Sxx Complex singularities
32P05 Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32-XX describing the type of problem)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)