Picard-Fuchs equations of special one-parameter families of invertible polynomials

2011/09/15 by Swantje Gährs, Gährs, Swantje
Mathematics · #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.1109.3462

Abstract

The thesis deals with calculating the Picard-Fuchs equation of special one-parameter families of invertible polynomials. In particular, for an invertible polynomial g(x₁,...,x_n) we consider the family f(x₁,...,x_n)=g(x₁,...,x_n)+s⋅∏ x_i, where s denotes the parameter. For the families of hypersurfaces defined by these polynomials, we compute the Picard-Fuchs equation, i.e. the ordinary differential equation which solutions are exactly the period integrals. For the proof of the exact appearance of the Picard-Fuchs equation we use a combinatorial version of the Griffiths-Dwork method and the theory of \GKZ systems. As consequences of our work and facts from the literature, we show the relation between the Picard-Fuchs equation, the Poincaré series and the monodromy in the space of period integrals.

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