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A.2.3 slim Groebner bases
The commandslimgb calls an implementation of an algorithm to compute
Groebner bases which is designed for keeping the polynomials slim (short
with small coefficients) during a Groebner basis computation. It provides,
in particular, a fast algorithm for computing Groebner bases over function
fields or over the rational numbers, but also in several other
cases. The algorithm which is still under developement was developed in
the diploma thesis of Michael Brickenstein. It has been published as
http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/35/paper_35_full.ps.gz.
In the example below (Groebner basis with respect to degree reverse
lexicographic ordering over function field) slimgb is much faster than
the std command.
ring r=(32003,u1, u2, u3, u4),(x1, x2, x3, x4, x5, x6, x7),dp; timer=1; ideal i= -x4*u3+x5*u2, x1*u3+2*x2*u1-2*x2*u2-2*x3*u3-u1*u4+u2*u4, -2*x1*x5+4*x4*x6+4*x5*x7+x1*u3-2*x4*u1-2*x4*u4-2*x6*u2-2*x7*u3+u1*u2+u2*u4, -x1*x5+x1*x7-x4*u1+x4*u2-x4*u4+x5*u3+x6*u1-x6*u2+x6*u4-x7*u3, -x1*x4+x1*u1-x5*u1+x5*u4, -2*x1*x3+x1*u3-2*x2*u4+u1*u4+u2*u4, x1^2*u3+x1*u1*u2-x1*u2^2-x1*u3^2-u1*u3*u4+u3*u4^2; i=slimgb(i); |
For detailed information and limitations see slimgb.