Complete intersection vanishing ideals on degenerate tori over finite fields

TL;DR

This paper investigates the algebraic properties of vanishing ideals on degenerate tori over finite fields, establishing a link with toric ideals and providing formulas for key invariants.

Contribution

It introduces a correspondence between vanishing ideals and toric ideals of numerical semigroups, preserving the complete intersection property and enabling new computational methods.

Findings

Established a correspondence between vanishing and toric ideals

Provided formulas for degree and regularity index based on numerical semigroup invariants

Showed algorithms can determine the complete intersection property

Abstract

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.