F5C: a variant of Faugere's F5 algorithm with reduced Groebner bases

TL;DR

F5C is an improved variant of Faugere's F5 algorithm that reduces intermediate Groebner bases to speed up computation and decrease polynomial reductions, with a generalized characterization theorem.

Contribution

F5C introduces reduced Groebner bases into F5, significantly enhancing efficiency and providing a generalized theorem for Groebner bases.

Findings

F5C terminates faster than F5.

F5C performs fewer polynomial reductions.

F5C considers fewer polynomials during computation.

Abstract

Faugere's F5 algorithm computes a Groebner basis incrementally, by computing a sequence of (non-reduced) Groebner bases. The authors describe a variant of F5, called F5C, that replaces each intermediate Groebner basis with its reduced Groebner basis. As a result, F5C considers fewer polynomials and performs substantially fewer polynomial reductions, so that it terminates more quickly. We also provide a generalization of Faugere's characterization theorem for Groebner bases.