## Modular algorithms for computing Gröbner bases.(English)Zbl 1046.13018

Summary: 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 the author presents two modular algorithms for computing a Gröbner basis for an ideal in $$\mathbb{Q}[x_1, \dots,x_\nu]$$ which extend the modular GCD algorithms. These algorithms improve upon previously proposed modular techniques for computing Gröbner bases in that the author tests primes before lifting, and also provides an algorithm for checking the result for correctness. A complete characterization of unlucky primes is also given. Finally, the author gives some preliminary timings which indicate that these modular algorithms can provide considerable time improvements in examples where intermediate coefficient growth is a problem.

### MSC:

 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 68W30 Symbolic computation and algebraic computation

### Keywords:

modular algorithms; unlucky primes; time improvements

### Software:

modwalk; modstd.lib; ffmodstd; Macaulay2; CoCoA
Full Text:

### References:

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