An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences

TL;DR

This paper introduces an efficient algorithm for computing leading monomials of minimal Groebner bases of generic sequences, leveraging structural properties and Hilbert series to improve performance.

Contribution

The paper presents a novel algorithm that bypasses polynomial reductions by exploiting conjectured properties of generic sequences, enhancing efficiency in Groebner basis computations.

Findings

Algorithm reduces computation time compared to traditional methods.

Memory usage is significantly decreased with the proposed approach.

Experimental results validate the efficiency and effectiveness of the algorithm.

Abstract

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties conjectured to hold for generic sequences-specifically, that their leading monomial ideals are weakly reverse lexicographic and that their Hilbert series follow a known closed-form expression. The algorithm incrementally constructs the set of leading monomials degree by degree by comparing Hilbert functions of monomial ideals with the expected Hilbert series of the input ideal. To enhance computational efficiency, we introduce several optimization techniques that progressively narrow the search space and reduce the number of divisibility checks required at each step. We also refine the loop termination condition using degree bounds, thereby avoiding unnecessary recomputation of Hilbert series. Experimental results confirm that the proposed method substantially reduces both computation time and memory usage compared to conventional Groebner basis computations for computing the leading monomials of a minimal Groebner basis of generic sequences.