On the Complexity of the F5 Gr\"obner basis Algorithm

TL;DR

This paper analyzes the complexity of Faugère's F5 algorithm for computing Gröbner bases, providing bounds on polynomial degrees and demonstrating its efficiency compared to traditional methods, especially for large degrees and many variables.

Contribution

It offers new bounds on the number of polynomials in Gröbner bases computed by F5 in generic cases and analyzes its structure to estimate complexity, showing its superior performance.

Findings

F5 outperforms row reduction of Macaulay matrices for moderate degrees.

The bounds on polynomial degrees are established for generic variable positions.

F5's efficiency grows exponentially with the number of variables for fixed degrees.

Abstract

We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree $d$ in a Gr\"obner basis computed by Faug\`ere's $F_5$ algorithm~(Fau02) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Gr\"obner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Gr\"obner bases with signatures that $F_5$ computes and use it to bound the complexity of the algorithm. Our estimates show that the version of~$F_5$ we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassen's multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.