factor out of an expression, a couple principal ideals, software?

Question

I have an expression, $f$, consisting of a few rational fractions of large multivariate numerators, $n1,\,n2,\ldots \in \mathbb{Q}[a1,a2,b1,b2;Q]$ and large multivariate denominators, $d1,\,d2,\ldots \in \mathbb{Q}[a1,a2,b1,b2;Q]$, in five indeterminates. Written succintly as $f:= \frac{n1}{d1} + \frac{n2}{d2} + \cdots$ (they are large though in a CAS program).

I can simplify it significantly by utilizing the fact that I have two `zeros' given by

$r1(a1,b1;Q)=0$ and $r2(a2,b2;Q)=0$.

I'm not too partial on the CAS, most is written in Maxima, but I would really like to stay with something like Maxima, Sage, Macaulay2, or SINGULAR. Of course I need a conceptual understanding first, which is where you smart people come in ;)

Answer 1

In Maple you can use simplify with side relations. The most common syntax is:

simplify(f, {r1,r2});

This will create a big rational expression and reduce the numerator and denominator separately using Groebner basis techniques. If you provide a monomial ordering, e.g.:

simplify(f, {r1,r2}, tdeg(a1,a2,b1,b2,Q));

Maple will minimize the largest monomial appearing in the result with respect to that ordering, again using Groebner bases. The third syntax is:

simplify(f, {r1,r2}, 'mindeg');

This will minimize the sum of the degrees of the numerator and denominator via direct search, which can become expensive for large inputs.