Computing the Radical of an Ideal in Positive Characteristic

Abstract

We propose a method for computing the radical of an arbitrary ideal in the polynomial ring in n variables over a perfect field of characteristic p > 0. In our method Buchberger's algorithm is performed once in n variables and a Gröbner basis conversion algorithm is performed at most n log p d times in 2n variables, where d is the maximum of total degrees of generators of the ideal and 3. Next we explain how to compute radicals over a finitely generated coefficient field over a field K, when we have a radical computation method over the field K. Thus we can compute radicals over any finitely generated field over a perfect field.

The Calculation of Radical Ideals in Positive Characteristic

“…But the ratios between the timings for different examples roughly correspond to those obtained in Matsumoto (2001).…”

“…Of course this ideal is already radical except in characteristic 2. The other test ideals are labelled E2, E3, L, M, 8 3 , and C. We do not reprint them here, since they can be found in Matsumoto (2001) or Caboara et al (1997). In order to obtain meaningful running times, the author implemented Matsumoto's algorithm in MAGMA, and compared the timings with the MAGMA implementation of the algorithm from this paper.…”

“…• Our timings for Matsumoto's algorithm are generally somewhat smaller than the ones given in Matsumoto (2001), except for L in characteristic 2. But the ratios between the timings for different examples roughly correspond to those obtained in Matsumoto (2001).…”

“…The entry "infeasible" in Table 1 signifies that after about 30 min of computing the algorithm was not even near completion. Every computation was attempted in the characteristics 2, 3, 5, 7, 11, 53, and 251, following the standard of Matsumoto (2001). If a characteristic does not appear in Table 1, this means that none of the algorithms was feasible.…”

“…While this paper was in the refereeing process, Matsumoto published an article (Matsumoto, 2001) which gives an entirely different algorithm for calculating the radical of an ideal in a polynomial ring in positive characteristic. Therefore a section has been added to this article which contains a brief discussion of Matsumoto's algorithm and some comparisons of performance.…”