Multigraded Hilbert function and toric complete intersection codes

TL;DR

This paper investigates the multigraded Hilbert function of zero-dimensional subschemes in toric varieties, providing explicit formulas and properties, and applies these results to evaluate codes on such subschemes.

Contribution

It introduces explicit formulas and properties for the multigraded Hilbert function of zero-dimensional complete intersections in toric varieties, with applications to coding theory.

Findings

Explicit formulas for $H_Y$ in toric varieties

Proved non-decreasing and stabilization properties of $H_Y$

Computed dimensions of evaluation codes on complete intersections

Abstract

Let $X$ be a complete $n$-dimensional simplicial toric variety with homogeneous coordinate ring $S$. We study the multigraded Hilbert function $H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit formulas and prove non-decreasing and stabilization properties of $H_Y$ when $Y$ is a $0$-dimensional complete intersection in $X$. We apply our results to computing the dimension of some evaluation codes on $0$-dimensional complete intersection in simplicial toric varieties.