On the complexity of the F5 Gröbner basis algorithm

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 (2002)...

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Bibliographic Details
Published in:Journal of symbolic computation 2015-09, Vol.70, p.49-70
Main Authors: Bardet, Magali, Faugère, Jean-Charles, Salvy, Bruno
Format: Article
Language:English
Subjects:
[formula omitted] algorithm
Complexity
Computer Science
Gröbner bases
Noether position
Regular sequences
Symbolic Computation
Online Access:Get full text
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Summary: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 (2002) 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.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2014.09.025