Hypergeometric periods for a tame polynomial

TL;DR

This paper studies the Gauss-Manin system associated with tame polynomials on smooth affine varieties, providing solutions to classical problems and establishing Hodge-theoretic properties similar to isolated hypersurface singularities.

Contribution

It offers a comprehensive analysis of the Gauss-Manin system for tame polynomials, solving the Birkhoff problem and proving Hodge-type results in this context.

Findings

Solution to the Birkhoff problem for the Gauss-Manin system

Hodge-type results analogous to isolated hypersurface singularities

Extension of classical singularity theory to tame polynomial cases

Abstract

We analyse the Gauss-Manin system of differential equations---and its Fourier transform---attached to regular functions satisfying a tameness assupmption on a smooth affine variety over C (e.g. tame polynomials on C^{n+1}). We give a solution to the Birkhoff problem and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities.