Algebraic differential equations associated to some polynomials

TL;DR

This paper computes the Gauss-Manin differential equations for polynomial periods with specific monomial structures and provides algebraic factorizations of these equations, advancing understanding of their algebraic properties.

Contribution

It introduces explicit computations of Gauss-Manin equations for polynomials with (n+2) monomials and presents two general algebraic factorization theorems for these differential equations.

Findings

Explicit Gauss-Manin differential equations for polynomials with (n+2) monomials.

Two general algebraic factorization theorems for these equations.

Enhanced understanding of the algebraic structure of polynomial period differential equations.

Abstract

We compute the Gauss-Manin differential equation for any period of a polynomial in \ $\C[x_{0},\dots, x_{n}]$ \ with \ $(n+2)$ \ monomials. We give two general factorizations theorem in the algebra \ $\C< z, (\frac{\partial}{\partial z})^{-1}>$ \ for such a differential equations.