The Degree and regularity of vanishing ideals of algebraic toric sets over finite fields

TL;DR

This paper investigates the algebraic properties of vanishing ideals associated with algebraic toric sets over finite fields, relating their invariants to graph structures, and applies findings to coding theory.

Contribution

It establishes connections between algebraic invariants of vanishing ideals and graph properties, computes degrees and regularity, and verifies the Eisenbud-Goto conjecture for certain graphs.

Findings

Computed degree and regularity for specific cases.

Proved the Eisenbud-Goto regularity conjecture for Hamiltonian bipartite graphs.

Provided bounds for the minimum distance of parameterized linear codes.

Abstract

Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.