Usage of modular techniques for efficient computation of ideal operations. (English) Zbl 1402.13026

Modular techniques are a powerful tool in computer algebra to improve the performance of the computations over the field of rationals to avoid the growth of intermediate coefficients. The main idea is to choose some lucky prime numbers, performing the computation modulo these primes and then reconstruct the the desired object by using the Chinese remainder theorem. In [J. Symb. Comput. 35, No. 4, 403–419 (2003; Zbl 1046.13018)], E. A. Arnold proposed an approach to test the luckiness of a given prime. For example, she defined a Hilbert lucky prime: A prime number \(p\) is called Hilbert lucky prime for an ideal if, modulo \(p\), the Hilbert function of the ideal is preserved. In the paper under review, the authors discuss different notions of a lucky prime which have been already proposed in the literature and define also a new notion of an affine Hilbert lucky prime. In addition, they apply the modular techniques for the computation of Gröbner bases, and some ideal operations.


13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)


Zbl 1046.13018


moddiq.lib; SINGULAR
Full Text: DOI


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