Let $R = \mathbf{k}[x_1,\ldots,x_n]$ be a polynomial ring over some field $\mathbf{k}$ (which can be $\mathbb{C}$ if that makes a difference) and $I$ some ideal of $R$ such that $R/I$ is finite-dimensional (as a $\mathbf{k}$-vector space). Let $B \subseteq R$ be some finite set of monomials. Is there an algorithmic procedure for deciding if $B$ is a $\mathbf{k}$-linear basis of $R/I$? If so, is it implemented in e.g. Sage or Macaulay2?
Apologies if this question is very trivial. I feel that Groebner basis theory must apply some way here but I cannot see it.
Compute a Groebner basis for the ideal. That gives you, among other things, a basis for the quotient made out of monomials.
Next, reduce the monomials in your B using the Groebner basis: this gives you their reduced expressions in the quotient. Check that these are linearly independent and have the correct number. Voilà.