Toric Difference Variety

TL;DR

This paper introduces the concept of toric difference varieties, providing multiple equivalent descriptions, establishing their connections with affine N[x]-semimodules, and presenting an algorithm to identify when a binomial difference ideal defines such a variety.

Contribution

It defines toric difference varieties and offers four equivalent characterizations, linking them to affine N[x]-semimodules and providing an algorithm for their identification.

Findings

Established correspondence between invariant subvarieties and faces of N[x]-submodules.

Proved the orbit-face correspondence for toric difference varieties.

Provided an algorithm to determine when a binomial difference ideal defines a toric difference variety.

Abstract

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the correspondence between the irreducible invariant difference subvarieties and the faces of the N[x]-submodules and the orbit-face correspondence. Finally, an algorithm is given to decide whether a binomial difference ideal represented by a Z[x]-lattice defines a toric difference variety.