ffmodstd

SINGULAR Library ffmodstd.lib: Groebner bases of ideals in polynomial rings over rational function fields. A library for computing a Groebner basis of an ideal in a polynomial ring over an algebraic function field Q(T):=Q(t_1,...,t_m) using modular methods and sparse multivariate rational interpolation, where the t_i are transcendental over Q. The idea is as follows: Given an ideal I in Q(T)[X], we map I to J via the map sending T to Tz:=(t_1z+s_1,..., t_mz+s_m) for a suitable point s in Q^m{(0,...,0)} and for some extra variable z so that J is an ideal in Q(Tz)[X]. For a suitable point b in Z^m{(0,...,0)}, we map J to K via the map sending (T,z) to (b,z), where b:=(b_1,...,b_m) (usually the b_i’s are distinct primes), so that K is an ideal in Q(z)[X]. For such a rational point b, we compute a Groebner basis G_b of K using modular algorithms [1] and univariate rational interpolation [2,7]. The procedure is repeated for many rational points b until their number is sufficiently large to recover the correct coeffcients in Q(T). Once we have these points, we obtain a set of polynomials G by applying the sparse multivariate rational interpolation algorithm from [4] coefficient-wise to the list of Groebner bases G_b in Q(z)[X], where this algorithm makes use of the following algorithms: univariate polynomial interpolation [2], univariate rational function reconstruction [7], and multivariate polynomial interpolation [3]. The last algorithm uses the well-known Berlekamp/Massey algorithm [5] and its early termination version [6]. The set G is then a Groebner basis of I with high probability.