Return explicit polynomial in .in_subalgebra

[schcs]

Problem Description

Given a polynomial f∈𝔽[x1,…,xn] which is an element of the subalgebra generated by f1,…,fk, calling f.in_subalgebra([f1,...,fk]) should returnh(t1,…,tk)such that f=h(f1,…,fk). I believe that this would be a useful feature.

Proposed Solution

The method is implemented in sage/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx. When algorithm = "groebner" is chosen, the polynomial z should be returned together with True. In case of False, 0 can be returned.

Alternatives Considered

I'm not sure about feasible alternatives.

Additional Information

No response

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

[user202729]

This is already possible as follows

sage: R.<x> = QQ[]
sage: S.<y> = QQ[]
sage: S.hom([x^2]).inverse_image(x^4)
y^2

(performance improvement might be possible)

On an unrelated note, for some reason the following is not implemented

x^4 in S.hom([x^2]).image()