Computing all border bases for ideals of points

TL;DR

This paper introduces two new term-ordering-free algorithms for computing all border bases of the vanishing ideal of a finite set of points, enhancing flexibility and efficiency in algebraic computations.

Contribution

The paper presents novel algorithms based on Buchberger-Möller and Farr-Gao methods that compute all border bases without relying on term orderings, with implementation and benchmarking.

Findings

Algorithms successfully compute all border bases for point ideals.

The methods are term ordering free, increasing flexibility.

Benchmarks demonstrate efficiency of the proposed algorithms.

Abstract

In this paper we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context two different approaches are discussed: based on the Buchberger-M\"oller Algorithm, we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr-Gao Algorithm for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.