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D.15.1 algemodstd_lib
- Library:
- algemodstd.lib
- Purpose:
- Groebner bases of ideals in polynomial rings
over algebraic number fields
- Authors:
- D.K. Boku [email protected]
W. Decker [email protected]
C. Fieker [email protected]
- Overview:
- 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.
- References:
- [1] E. A. Arnold: Modular algorithms for computing Groebner bases.
J. Symb. Comp. 35, 403-419 (2003).
Procedures:
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