CoCoA » CoCoALib

For min polys it might be possible to work with a sub-ideal in a subring. For example,

use QQ[x,y,z,t];
I := ideal(x^2-2, y^3-3, z^5-5);
f := x+y;
MinPolyQuot(f,I,x);

In the call to MinPolyQuot we could work in the subring QQ[x,y] with the ideal I2 := ideal(x^2-2, y^3-3).

One way to do this would be to build a graph: for each indet there is a node, and
there is an edge between two "indets" if they both appear in some generator of I (or if we have a reduced GBasis, then we use the GBasis elements instead of the gens). The "active" indets are then those in the connected component of the graph corresponding to the indets in the polynomial f. In the above example we see that
only x and y are "active": so we can work with only those gens which involve
just x and/or y, and ignore all the other gens.

Mathemetically we could also work with an elimination ideal, but computing it might be too costly... but perhaps the elim ideal is trivial in the case that the graph described above is conveniently disconnected.