A more or less interesting problem I have encountered during one of the courses I took was the following:
(This is a toric variety associated to some semi-lattice S)
Now we are given some element of our vectorspace and want to check whether it is actually in V. This is not as simple as it looks in the first place:
So far, I've not come up with a general method for deciding whether such a point is contained in V or not. For some special cases it is not difficult to prove that checking the condition for only the basis of M and some (finitely many) more elements of M is enough, although I don't know if in general, checking a finite number of equations suffices.
