Groebner basis and singular locus of Lauricella's hypergeometric differential equations

TL;DR

This paper computes Groebner bases for Lauricella's hypergeometric differential equations to identify their characteristic varieties and singular loci, advancing understanding of their algebraic and geometric properties.

Contribution

It introduces explicit Groebner bases for Lauricella's equations and uses them to analyze their singularities and characteristic varieties.

Findings

Groebner bases for $I_A(m), I_B(m), I_C(m)$ derived

Characteristic varieties and singular loci determined

Enhanced understanding of the equations' algebraic structure

Abstract

We derive Groebner bases for Lauricella's hypergeometric differential equations $I_A(m), I_B(m), I_C(m)$ with respect to a monomial order. By using these Groebner bases, we determine characteristic varieties and the singular loci of $I_A(m), I_B(m), I_C(m)$.