CoCoA » CoCoALib

I have just started working on an implementation of isomorphic-to-subring. For example, given QQ[x,y,z] it should be
possible to create an isomorphic copy of the subring generated by z: effectively a poly ring QQ[#1] is created together with two homomorphisms. One hom sends x |--> 0 and y |--> 0 but z |--> #1; the reverse hom sends #1 |--> z.

The question is what term-ordering to put on the newly created ring. The default is to have an ordering which is the restriction of the term-ordering in the big ring. However there seems to be a "problem" with gradings.

Suppose the grading of z is (0,1); this means that a compatible grading on Q[#1] also has to be (0,1) but the ring has only 1 indet, and we have supposed that the term-ordering is compatible with the grading and the term-ordering is given by a square matrix... not possible in this case! What to do? Just forget the grading?

My current preference is to allow gradings with negative weights, and possibly linearly dependent weight matrices.
This would mean that the internal repr of the grading differs from that which the user expects to see, but the two are in 1--1 correspondence (via a matrix, not nec invertible!).

Taking the example at the end of comment 1 (#832#note-1). The internal grading would be [1] for z, but this would be mapped into ZZ^2 by multiplying by the 2x1 matrix ColMat([0,1]). This does make functions such as wdeg more complicated, and slower, but that may not matter. Also there would have be functions to map from the grading the user expects to the internal representation, and vice versa.