Paper Info
Title
A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains.
Abstract
Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).
Year
DOI
Venue
2020
10.1007/s11786-019-00432-5
Mathematics in Computer Science
Keywords
DocType
Volume
Algorithms, Gröbner bases, Signature-based algorithms, Polynomials over rings, Principal ideal domains, 13P10
Journal
14
Issue
ISSN
Citations 
2
1661-8270
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Maria Francis103.72
Thibaut Verron200.34