Rational points of lattice ideals on a toric variety and toric codes

TL;DR

This paper establishes a method to compute the number of rational points of subgroups in toric varieties over finite fields using Smith normal form, and applies this to analyze parameters of generalized toric codes.

Contribution

It generalizes the computation of rational points for lattice ideals on toric varieties and connects subgroup classification with lattice ideals, with applications to toric codes.

Findings

Number of rational points computed via Smith normal form.

Established a Nullstellensatz type theorem over finite fields.

Determined parameters of generalized toric codes on Hirzebruch surfaces.

Abstract

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature.