Minimal Generating Sets of Lattice Ideals

Abstract

Let $L\subset {\mathbb{Z}}^n$ be a lattice and $I_L=⟨x^{\mathbf{u}}-x^{\mathbf{v}}:\mathbf{u}-\mathbf{v}\in L⟩$ be the corresponding lattice ideal in $\mathbb{k}[x_1,\dots ,x_n]$, where $\mathbb{k}$ is a field. In this paper we describe minimal binomial generating sets of $I_L$ and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of $I_L$. As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices.

v4: the title is changed, a few proofs simplified and one example added (Example 4.9); to appear in Collectanea Math

References in corpus (3)

• Higher Lawrence configurations
• Decompositions of commutative monoid congruences and binomial ideals
• Complete intersections in binomial and lattice ideals

Cited by in corpus (1)

• On the relative size of toric bases