Finite subsets of projective space, and their ideals

TL;DR

This paper investigates the relationship between finite sets of rational points in projective space and their ideals, establishing a connection between the set size and axes in a specific monomial set, and provides an algorithm for computing their Gröbner bases.

Contribution

It introduces a novel relationship between the size of point sets and axes in monomial sets, along with an algorithm for Gröbner basis construction of their ideals.

Findings

Size of point set equals number of axes in monomial set

Algorithm for Gröbner basis construction provided

Enhanced understanding of ideals of finite projective point sets

Abstract

Let $\mathscr{A}$ be a finite set of closed rational points in projective space, let $\mathscr{I}$ be the vanishing ideal of $\mathscr{A}$, and let $\mathscr{D}(\mathscr{A})$ be the set of exponents of those monomials which do not occur as leading monomials of elements of $\mathscr{I}$. We show that the size of $\mathscr{A}$ equals the number of axes contained in $\mathscr{D}(\mathscr{A})$. Furthermore, we present an algorithm for the construction of the Gr\"obner basis of $\mathscr{I}(\mathscr{A})$, hence also of $\mathscr{D}(\mathscr{A})$.