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Working with a polynomial ring $R$, if I have an ideal generated by a set of polynomials $f_i$ (taking two for concreteness), I can obtain a Groebner basis $g_j$ by the commands

I = R.ideal([f1, f2])
g = I.groebner_basis()

What I want to do is express the Groebner basis in terms of the original basis, i.e. $g_j = \sum_i a_{ij} f_i$. How can I accomplish this in Sage? My understanding is that such an expression is calculated implicitly when the Groebner basis is calculated, so perhaps it is possible to recover it directly from the algorithm.

That's much appreciated, thank you. One final question on the extra variables that you add, just to make sure I'm understanding - the reduce function does the reduction in a reduced Groeber basis, and this procedure should cause problems if the original basis isn't a Groeber basis, correct? It seems that if the original basis isn't a Groebner basis, we get higher powers of c1 and c2 in the remainder (although there's always only one unique power of c1 and c2 involved in the remainder). Taking the coefficients corresponding to these higher powers of c1 and c2 seem to still give sensible answers for the quotients in the cases that I've considered, but I haven't been able to convince myself that this is always the correct thing to do.

Thanks! This is very helpful. Do you if the division algorithm they reference (or in general, division by several polynomials) is implemented in Sage?

I have a working example shown below to help explain my question.

R.<x> = GF(2)[];
I = R.ideal([x^2 - 1])
S.<u> = R.quotient_ring(I)
f = x^2 + x
g = x
print(f.quo_rem(g))
f = u^2 + u
g = u
print(f.quo_rem(g))

The first quo_rem call works fine, but the second does not. Is polynomial division not supported for quotient rings, or am I calling it the wrong way? I'm specifically interested in doing this for multivariate polynomials, but I can't seem to get it to work for a single variable either.

Related to the question, I am interested in understanding this division in quotient rings better. If the operation isn't supported in sage, any sort of references/algorithms on how to do such a calculation would be very helpful and appreciated.