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D.15.1 algemodstd_lib

Groebner bases of ideals in polynomial rings over algebraic number fields
D.K. Boku boku@mathematik.uni-kl.de
W. Decker decker@mathematik.uni-kl.de
C. Fieker fieker@mathematik.uni-kl.de

A library for computing the Groebner basis of an ideal in the polynomial ring over an algebraic number field Q(t) using the modular methods, where t is algebraic over the field of rational numbers Q. For the case Q(t) = Q, the procedure is inspired by Arnold [1]. This idea is then extended to the case t not in Q using factorization as follows: Let f be the minimal polynomial of t.
For I, I' ideals in Q(t)[X], Q[X,t]/<f> respectively, we map I to I' via the map sending t to t + <f>.
We first choose a prime p such that f has at least two factors in characteristic p and add each factor f_i to I' to obtain the ideal J'_i = I' + <f_i>. We then compute a standard basis G'_i of J'_i for each i and combine the G'_i to G_p (a standard basis of I'_p) using chinese remaindering for polynomials. The procedure is repeated for many primes p, where we compute the G_p in parallel until the number of primes is sufficiently large to recover the correct standard basis G' of I'. Finally, by mapping G' back to Q(t)[X], a standard basis G of I is obtained.

[1] E. A. Arnold: Modular algorithms for computing Groebner bases. J. Symb. Comp. 35, 403-419 (2003).


D.15.1.1 chinrempoly  chinese remaindering for polynomials
D.15.1.2 algemodStd  standard basis of I over algebraic number field using modular methods