2013/12/05 by Magali Bardet, Bardet, Magali, Jean‐Charles Faugère +3 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Computer and information sciences #Polynomial and algebraic computation #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.1312.1655
openalex publication_date 2013/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the complexity of Gröbner 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 Gröbner basis computed by Faugère's F5 algorithm~(Fau02) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Gröbner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Gröbner bases with signatures that F5 computes and use it to bound the complexity of the algorithm. Our estimates show that the version of~F5 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.