A Two-Dimensional Improvement for Farr-Gao Algorithm

TL;DR

This paper introduces a two-dimensional preprocessing enhancement for the Farr-Gao algorithm, significantly improving its efficiency in computing reduced Gr"obner bases for finite point sets.

Contribution

It presents a novel preprocessing strategy for tower subsets that accelerates the Farr-Gao algorithm's performance in Gr"obner basis computations.

Findings

Preprocessed Farr-Gao algorithm is more efficient than the classical version.

Experimental results demonstrate improved computational speed.

The method effectively handles tower subsets of point sets.

Abstract

Farr-Gao algorithm is a state-of-the-art algorithm for reduced Gr\"{o}bner bases of vanishing ideals of finite points, which has been implemented in Maple$^\circledR$ as a build-in command. In this paper, we present a two-dimensional improvement for it that employs a preprocessing strategy for computing reduced Gr\"{o}bner bases associated with tower subsets of given point sets. Experimental results show that the preprocessed Farr-Gao algorithm is more efficient than the classical one.