Regularity and algebraic properties of certain lattice ideals

TL;DR

This paper investigates the algebraic properties and regularity of lattice ideals, establishing a map between graded lattice ideals in different polynomial rings, and explores applications over finite fields and projective spaces.

Contribution

It introduces a map between lattice ideals in differently graded rings, preserving key properties, and provides new formulas and constructions for regularity and vanishing ideals over finite fields.

Findings

The map I --> I~ preserves complete intersection property and regularity.

Derived a formula for the regularity of vanishing ideals of degenerate tori over finite fields.

Characterized when the vanishing ideal of a subset in projective space is a lattice ideal of dimension 1.

Abstract

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields, with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartie graph in terms of the regularities of the vanishing ideals of the blocks of the graph.