Refined algorithms to compute syzygies

Based on Schreyer’s classical algorithm, we present a refined algorithm for the computation of syzygies. The main ideas 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''. 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. Third, we may additionally cache some partial results and reuse them at the remaining steps.

Altogether, these improvements allow us to tackle previously unsolvable computational problems such as the Prym-Green conjecture for curves of genus 24.