Computing syzygies by Faugère's \(\mathbb{F}_{5}\) algorithm (Q2430791)

The authors introduce a new efficient algorithm for computing a set of generators for the first syzygies on a sequence \(F\) of homogeneous polynomials. For this, they extend a given sequence of polynomials to a Gröbner basis using \textit{J.-C. Faugère}'s \(\mathbb{F}_5\) algorithm [ISSAC 2002. Proceedings of the 2002 international symposium on symbolic and algebraic computation, Lille, France, July 07--10, 2002. New York, NY: ACM Press. 75--83 (2002; Zbl 1072.68664)]. They show that if one keeps all the reductions to zero during this computation, then at termination, by adding principal syzygies, one obtains a basis for the module of syzygies on the input polynomials. Precisely, the main theorem of this paper states that the module \(Syz(F)\) is equal to the union of the principal syzygies on \(F\) and the syzygies corresponding to the reductions to zero of \(\mathbb{F}_5\) if the algorithm is executed on \(F\) with respect to a monomial ordering. The authors have implemented this algorithm in the computer algebra system \texttt{Magma}, and they have evaluated its performance via some examples.