Refined Algorithms to Compute Syzygies

TL;DR

This paper introduces two optimized algorithms, LiftHybrid and LiftTree, that improve the efficiency of computing syzygies by reducing unnecessary calculations and reusing partial results based on Schreyer's algorithm.

Contribution

The paper presents two refined algorithms for syzygy computation that significantly reduce computational complexity and improve efficiency over traditional methods.

Findings

LiftHybrid reduces monomial comparisons by skipping lower order terms.

LiftTree caches partial results for reuse, enhancing efficiency.

Both algorithms outperform classical Schreyer's algorithm in computational tests.

Abstract

Based on Schreyer's algorithm (Schreyer, 1980, 1991; Berkesch and Schreyer, 2014), we present two refined algorithms for the computation of syzygies. The two main ideas of the first algorithm, called LiftHybrid, are the following: First, we may leave out certain terms of module elements during the computation which do not contribute to the result. These terms are called "lower order terms", see Definition 4.2. Second, we do not need to order the remaining terms of these module elements during the computation. This significantly reduces the number of monomial comparisons for the arithmetic operations. For the second algorithm, called LiftTree, we additionally cache some partial results and reuse them at the remaining steps.