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Journal of Symbolic Computation 35 (2003) 403419 www.elsevier.com/locate/jsc

Summary: Journal of Symbolic Computation 35 (2003) 403­419
Modular algorithms for computing Gršobner bases
Elizabeth A. Arnold
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
Received 5 October 2000; accepted 14 March 2002
Intermediate coefficient swell is a well-known difficulty with Buchberger's algorithm for
computing Gršobner bases over the rational numbers. p-Adic and modular methods have been
successful in limiting intermediate coefficient growth in other computations, and in particular in
the Euclidian algorithm for computing the greatest common divisor (GCD) of polynomials in one
variable. In this paper we present two modular algorithms for computing a Gršobner basis for
an ideal in Q[x1, . . . , x] which extend the modular GCD algorithms. These algorithms improve
upon previously proposed modular techniques for computing Gršobner bases in that we test primes
before lifting, and also provide an algorithm for checking the result for correctness. A complete
characterization of unlucky primes is also given. Finally, we give some preliminary timings which
indicate that these modular algorithms can provide considerable time improvements in examples
where intermediate coefficient growth is a problem. © 2003 Published by Elsevier Science Ltd.
1. Introduction
Intermediate coefficient swell is a notorious difficulty of Buchberger's algorithm for


Source: Arnold, Elizabeth A. - Department of Mathematics and Statistics, James Madison University


Collections: Mathematics