The vanishing ideal of a finite set of closed points in affine space

TL;DR

This paper presents a new inductive method to compute a Gröbner basis for the ideal of polynomials vanishing on a finite set of points in affine space, avoiding linear system solutions.

Contribution

It introduces an alternative to the Buchberger–Möller algorithm, determining leading terms and basis elements through induction and interpolation rather than linear equations.

Findings

Provides a Gröbner basis for the ideal of points

Determines leading terms without solving linear systems

Uses induction over the dimension of affine space

Abstract

Given a finite set of closed rational points of affine space over a field, we give a Gr\"obner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger--M\"oller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gr\"obner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.