Toric ideals for high Veronese subrings of toric algebras

TL;DR

This paper proves that high Veronese subrings of toric algebras have quadratic Gr"obner bases, providing explicit bounds and generalizing to standard graded rings, which advances understanding of their algebraic structure.

Contribution

It establishes that sufficiently high Veronese subrings of toric algebras have quadratic Gr"obner bases, with explicit bounds improving previous results.

Findings

Existence of quadratic Gr"obner bases for high Veronese subrings

Explicit lower bounds on the degree for quadratic Gr"obner bases

Comparison showing improved bounds over prior work

Abstract

We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gr\"obner basis consisting of binomials. More generally, we prove that the defining ideal of a sufficiently high Veronese subring of a standard graded ring admits a quadratic Gr\"obner basis. We give a lower bound on $d$ such that the defining ideal of $d$-th Veronese subring admits a quadratic Gr\"obner basis. Eisenbud--Reeves--Totaro stated the same theorem without a proof with some lower bound on $d$. In many cases, our lower bound is less than Eisenbud--Reeves--Totaro's lower bound.