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Monodromy of dual invertible polynomials. (English) Zbl 1257.32028

Summary: A generalization of Arnold’s strange duality to invertible polynomials in three variables by the first author and A. Takahashi [Compos. Math. 147, No. 5, 1413–1433 (2011; Zbl 1238.14029)] includes the following relation. For some invertible polynomials \(f\) the Saito dual of the reduced monodromy zeta function of \(f\) coincides with a formal “root” of the reduced monodromy zeta function of its Berglund-Hübsch transpose \(f^{T}\). Here we give a geometric interpretation of “roots” of the monodromy zeta function and generalize the above relation to all non-degenerate invertible polynomials in three variables and to some polynomials in an arbitrary number of variables in a form including “roots” of the monodromy zeta functions both of \(f\) and \(f^{T}\).

MSC:

32S05 Local complex singularities
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14J33 Mirror symmetry (algebro-geometric aspects)

Citations:

Zbl 1238.14029
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