I would like to define the $K$-vector space generated by a finite basis of multivariate polynomials (in $n$ variables, over the field $K$). My goal is then to find out the components of any element in this vector space w.r.t this basis. This is not a Gröbner basis problem since the coefficients are not polynomials. In my application, $K$ is finite.
It is easy to do in the univariate case, but I don't see how to proceed in general. Should I define this space as a subspace of a larger finite dimensional vector space over $K$ (for example, polynomials of some bounded total degree) ?
Thank you for your help :)