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

TL;DR

This paper computes the Picard-Fuchs equations for specific one-parameter families of invertible polynomials, linking these equations to period integrals, Poincaré series, and monodromy using combinatorial and GKZ methods.

Contribution

It introduces a combinatorial approach combined with GKZ systems to explicitly compute Picard-Fuchs equations for these polynomial families.

Findings

Derived explicit Picard-Fuchs equations for the families

Established the relation between Picard-Fuchs equations and monodromy

Connected the equations to Poincaré series and period integrals

Abstract

The thesis deals with calculating the Picard-Fuchs equation of special one-parameter families of invertible polynomials. In particular, for an invertible polynomial $g(x_1,...,x_n)$ we consider the family $f(x_1,...,x_n)=g(x_1,...,x_n)+s\cdot\prod x_i$, 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 Poincar\'{e} series and the monodromy in the space of period integrals.