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Computing the degree of a lattice ideal of dimension one

Hiram H. Lopez, Rafael H. Villarreal

TL;DR

The paper addresses the problem of computing the degree of a graded lattice ideal of dimension one. It proves that the degree equals the order of the torsion subgroup of the quotient Z^s/L, and that, if L is generated by the rows of an integral matrix, the degree equals the product of the Smith normal form invariant factors, offering a linear-algebraic route to computation. It also provides a geometric interpretation via relative volume and demonstrates practical efficiency, including applications to vanishing ideals over finite fields and coding theory. Several open questions are discussed, notably connections to graph invariants like sandpile groups and extensions to higher-dimensional lattice ideals.

Abstract

We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.

Computing the degree of a lattice ideal of dimension one

TL;DR

The paper addresses the problem of computing the degree of a graded lattice ideal of dimension one. It proves that the degree equals the order of the torsion subgroup of the quotient Z^s/L, and that, if L is generated by the rows of an integral matrix, the degree equals the product of the Smith normal form invariant factors, offering a linear-algebraic route to computation. It also provides a geometric interpretation via relative volume and demonstrates practical efficiency, including applications to vanishing ideals over finite fields and coding theory. Several open questions are discussed, notably connections to graph invariants like sandpile groups and extensions to higher-dimensional lattice ideals.

Abstract

We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.

Paper Structure

This paper contains 6 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Hilbert functions and the degree
  3. The degree of a lattice ring
  4. Computing some examples
  5. Vanishing ideals over finite fields
  6. Problems and related results

Key Result

Lemma 2.1

(a) If $S_i=I_i$ for some $i\geq 1$, then $S_d=I_d$ for all $d\geq i$. (b) If $\dim\, S/I\geq 1$, then $H_I(i)>0$ for $i\geq 0$.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • ...and 35 more