Efficiently computing Groebner bases of ideals of points

TL;DR

This paper introduces an optimized algorithm for computing Groebner bases of point ideals, significantly reducing computation time when the number of points is less than the number of variables, with applications in molecular biology.

Contribution

The paper presents a new algorithm that identifies essential variables and uses PLU decompositions to improve efficiency over standard methods for specific cases.

Findings

The new algorithm is an order of magnitude faster for large variable counts.

Implementation results confirm the theoretical speedup.

The approach is motivated by applications in molecular biology network reconstruction.

Abstract

We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Moeller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Moeller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Groebner bases to the problem of network reconstruction in molecular biology.