### D.4.22 nfmodsyz_lib

Library:
  nfmodsyz.lib
Purpose:
  Syzygy modules of submodules of free modules 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 syzygy module of a given submodule I in a polynomial ring over an algebraic number field Q(t), where t is an algebraic number, using modular methods. For the case Q(t)=Q, that is, where t is an element of Q, we compute, following [1], the syzygy module of I as follows: For a submodule I of A^m with A = Q[X], we first choose a sufficiently large set of primes P and compute the reduced Groebner basis of the syzygy module of I_p, for each p in P, in parallel. We then use the Chinese remainder algorithm and rational reconstruction to obtain the syzygy module of I over Q. For the case where t is not in Q, we compute, following [2], the syzygy module of I as follows:
  Let f be the minimal polynomial of t. For a submodule I in A^m with A = Q(t)[X], we map I to a submodule I' in A^m with A = (Q[t]/<f>)[X] 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. 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 the reduced Groebner bases G'_i of the syzygy modules of I'_{i,p} over F_p[t]/<f_{i,p}> and combine the G'_i to G_p (the syzygy module of I'_p) using chinese remaindering for polynomials. As described in [2], 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 generating set for the syzygy module G' of I' which is, considered over Q(t), also a generating set for the syzygy module of I.

References:
  [1] E. A. Arnold: Modular algorithms for computing Groebner bases. J. Symb. Comp. 35, 403-419 (2003).
  [2] D. Boku, W. Decker, C. Fieker, and A. Steenpass. Groebner bases over algebraic number fields. In: Proceedings of the 2015 International Workshop on Parallel Symb. Comp. PASCO'15, pages 16-24 (2015).

Procedures:

> D.4.22.1 nfmodSyz |  | syzygy module of I over algebraic number field using modular methods

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