Toric ideals for high Veronese subrings of toric algebras

We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gröbner 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 Gröbner basis. We give a lower bound on d such that the defining ideal of dth Veronese subring admits a quadratic Gröbner 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. 2013