Vanishing Ideals Of Affine Sets Parameterized By Odd Cycles

TL;DR

This paper provides explicit algebraic descriptions and formulas for the vanishing ideals of affine sets parameterized by odd cycles over finite fields, including their Gr"obner bases, regularity, and Hilbert functions.

Contribution

It introduces explicit Gr"obner bases and formulas for the regularity and Hilbert functions of vanishing ideals of affine sets parameterized by odd cycles.

Findings

Explicit Gr"obner basis for I(X*)

Formula for the regularity of I(X*)

Hilbert function expressed as a combination of degenerate torus Hilbert functions

Abstract

Let K be a finite field. Let X* be a subset of the affine space Kn, which is parameterized by odd cycles. In this paper we give an explicit Gr\"obner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.