
D.15.20 nfmodsyz_lib
 Library:
 nfmodsyz.lib
 Purpose:
 Syzygy modules of submodules of free modules over
algebraic number fields
 Authors:
 D.K. Boku boku@mathematik.unikl.de
W. Decker decker@mathematik.unikl.de
C. Fieker fieker@mathematik.unikl.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, 403419 (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 1624 (2015).
Procedures:
