Modular algorithms for computing Gröbner bases

Intermediate coefficient swell is a well-known difficulty with Buchberger’s algorithm for computing Gröbner 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 Gröbner basis for an ideal in Q[x1, . . . , xν ] which extend the modular GCD algorithms. These algorithms improve upon previously proposed modular techniques for computing Gröbner 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.