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Bernstein polynomials and spectral numbers for linear free divisors. (English) Zbl 1221.34237
By a classical result due to B. Malgrange, one can relate the eigenvalues of the monodromy acting on the Milnor fibre of an isolated hypersurface singularity to the roots of the Bernstein polynomial associated to the defining equation. In this article the author studies Bernstein polynomials of the defining equation \(h\) of an arbitrary reductive linear free divisor \(D\) and is able to prove a similar result in this case. First, he recalls some results concerning linear free divisors given in [I. D. Gregorio, D. Mond and C. Sevenheck, Compos. Math. 145, 5, 1305–1350 (2009; Zbl 1238.32022)] and Bernstein polynomials associated to functions and \(\mathcal{D}\)-modules. In the main result of the article, he compares the roots of the Bernstein polynomial associated to \(h\) to the roots of the Bernstein polynomial of a restriction of the logarithmic extension of the family of Gauß-Manin systems, which can be associated to a generic section of \(D\). The introduction of a suitable logarithmic Brieskorn lattice associated to \(h\) leads to an analouge of Malgrange’s result. Finally, he proves a symmetry property concerning the spectrum of this lattice at infinity, which is also similar to the case of an isolated hypersurface singularity, computes the Bernstein polynomials for some examples of linear free divisors and conjectures another symmetry property for the spectrum of the logarithmic Brieskorn lattice.

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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