The vanishing ideal of a finite set of points with multiplicity structures

TL;DR

This paper introduces an algorithm to compute the reduced Groebner basis of the vanishing ideal for finite point sets with multiplicities, revealing geometric relations and employing a novel ideal intersection method based on the Extended Euclidean Algorithm.

Contribution

The paper presents a new algorithm that efficiently computes the Groebner basis for point sets with multiplicities, linking geometric configurations to algebraic structures.

Findings

Algorithm successfully computes Groebner bases for complex point sets

Reveals geometric relationships between point positions and ideal structures

Uses a novel ideal intersection method based on the Extended Euclidean Algorithm

Abstract

Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method discloses the essential geometric connection between the relative position of the points with multiplicity structures and the quotient basis of the vanishing ideal, so we will explicitly know the set of leading terms of elements of I. We split the problem into several smaller ones which can be solved by induction over variables and then use our new algorithm for intersection of ideals to compute the result of the original problem. The new algorithm for intersection of ideals is mainly based on the Extended Euclidean Algorithm.