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D.4.9 ffmodstd_lib
- Library:
- ffmodstd.lib
- Purpose:
- Groebner bases of ideals in polynomial rings
over rational function fields
- Authors:
- D.K. Boku [email protected]
W. Decker [email protected]
C. Fieker [email protected]
A. Steenpass [email protected]
- Overview:
- 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], where prime numbers are replaced by maximal ideals of the form
<z-z_i>, and univariate rational interpolation [2,7]. Note that since Q[z]/<z-z_i> = Q
we also use (if required) modular algorithms [1] over Q. The procedure is repeated for
many rational points b until their number is sufficiently large to recover the correct
coefficients 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.
- References:
- [1] E. A. Arnold: Modular algorithms for computing Groebner bases.
J. Symb. Comput. 35, 403-419 (2003).
[2] R. L. Burden and J. D. Faires: Numerical analysis. 9th ed. (1993).
[3] M. Ben-Or and P. Tiwari: A deterministic algorithm for sparse multivariate
polynomial interpolation. Proc. of the 20th Annual ACM Symposium on
Theory of Computing, 301-309 (1988).
[4] A. Cuyt and W.-s. Lee: Sparse interpolation of multivariate rational functions.
Theor. Comput. Sci. 412, 1445-1456 (2011).
[5] E. Kaltofen and W.-s. Lee: Early termination in sparse interpolation algorithms.
J. Symb. Comput. 36, 365-400 (2003).
[6] E. Kaltofen, W.-s. Lee and A. A. Lobo: Early termination in Ben-Or/Tiwari
sparse interpolation and a hybrid of Zippel's algorithm. Proc. ISSAC
(ISSAC '00), 192-201 (2000).
[7] K. Sara and M. Monagan: Fast Rational Function Reconstruction. Proc. ISSAC
(ISSAC '06), 184-190 (2006).
Procedures:
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