### D.4.21 nfmodstd_lib

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

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.
  The procedure also works if the input is a module. For this, we consider the rings A = Q(t)[X] and A' = (Q[t]/<f>)[X]. For submodules I, I' in A^m, A'^m, respectively, we map I to I' via the map sending t to t + <f>. As above, we first choose a prime p such that f has at least two factors in characteristic p. For each factor f_{i,p} of f_p := (f mod p), we set I'_{i,p} := (I'_p mod f_{i,p}). We then compute a standard basis G'_i of I'_{i,p} over F_p[t]/<f_{i,p}> 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 as described above and we finally obtain a standard basis of I.

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

Procedures:

> D.4.21.1 chinrempoly |  | chinese remaindering for polynomials
> D.4.21.2 nfmodStd |  | standard basis of I over algebraic number field using modular methods

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