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. This was proved by Eisenbud–Reeves–Totaro in the case where coordinates are generic. Our proof does not need coordinate transformations. Note that a change of coordinates does not preserve the property that an ideal is generated by binomials in general. Notation: In this paper, we denote by N = {0, 1, 2, 3, . . .} the set of non-negative integers. For a multi-index a = (a1, . . . , ar) ∈ N r and variables x = x1, . . . , xr, we write x = x1 1 · · ·x ar r and |a| = a1 + · · ·+ ar. For a given positive integer s and d, we set Nsd = {a = (a1, . . . , as) ∈ N s | |a| = d}. We denote by ei the vector with unity in the i-th position and zeros elsewhere.