How to find a $k$-basis for a quotient space of a polynomial ring by a subspace?

Question

My question is "How to find a $k$-basis for a quotient space of a polynomial ring $R$ over $k$ by a subspace $V$?"

Here the point is quotient by a subspace, not by an ideal.

In Singular, there is a command "kbase".

But this command only computes the case when $V$ is an ideal.

Though it's not exactly what I want, I'd like to refer to the implementation. Can I see the source file itself? Or anybody can roughly explain it?

Does it have something to do with "Groebner basis"?(I'm asking because "kbase" command requires the Groebner basis.)

Thanks in advance.

Answer 1

Find a supplementary subspace $W$ to $V$ in $R$, and find a basis of $W$. Then the images of elements of this basis form a basis for the quotient vector space over $k$. Maybe there's an algorithm of basis completion, which gives you automatically basis of $W$ ?