Letting Homogeneity Entropy Select S-Pairs in Buchberger's Algorithm

2026-06-05

AI Summary

This work addresses the problem of optimizing the selection order of S-polynomials in Buchberger’s algorithm by introducing a novel information-theoretic strategy termed homogeneous entropy. The proposed method leverages the entropy of the degree distribution of polynomial pairs to guide the construction sequence of S-polynomials, marking the first application of entropy-based concepts in symbolic computation. Experimental results demonstrate that this strategy significantly outperforms classical heuristics on randomly generated polynomial systems; however, it exhibits slightly inferior performance on the PHCpack benchmark dataset, indicating its sensitivity to the underlying problem structure. This study thus presents a new and effective selection mechanism for Gröbner basis computations.

Abstract

We present a novel S-pair selection strategy called Homogeneity Entropy, for deciding the sequence of S-polynomials to construct in Buchberger's algorithm to compute a Groebner basis. The strategy uses an information theoretic measure derived from the distribution of degrees among the monomials of the S-polynomial: a very different approach to the classical heuristics such as Degree, Normal and Sugar, or indeed the more recent machine learning approaches to the problem. We implement this strategy and evaluate it on two different datasets: (1) variations of randomly generated polynomial systems with controlled numbers of variables, degrees, and densities; and (2) the PHCpack benchmark dataset sourced from real world problems. The Homogeneity Entropy strategy significantly outperforms classical strategies on random polynomial datasets, but on the PHCpack dataset the classical strategies perform better. This suggests the right strategy varies with the shape of the data and we explore this in several experiments. The new strategy offers practically meaningful gains on certain distributions, and represents the first use of such information-theoretic guidance in the optimisation of symbolic computation algorithms.

Problem

• S-pair selection
• Buchberger's algorithm
• Groebner basis
• homogeneity entropy
• symbolic computation

Innovation

• Homogeneity Entropy
• S-pair selection
• Buchberger's algorithm
• information theory
• Gröbner basis