Consider the ideal $I\subseteq S[x,y,z]$ where $S$ is some field of characteristic 0 (probably any field will do) and $I=\langle x^9-y^4z^4,y^9-x^5z^4,z^8-x^4y^5,x^6\rangle$. Notice that because the lone monomial in $I$ also divides half of one of the binomials, we have that $I=\langle y^9-x^5z^4,z^8-x^4y^5,x^6,y^4z^4\rangle$. What I am looking for is some sort of computational tool that will catch that division and tell me about the existence of this 'simpler' generating set. I was thinking that I'd have to use some sort of Groebner bases trick, but if the term order (in this case) decreed that $y^4z^4 > x^9$, then I can't imagine that the division would get noticed. So, in general, what I am looking for is an invariant on an ideal that says "if you write this ideal using as many monomials as possible whilst maintaining a finite minimal generating set, you have such and such many monomials". Any ideas out there? Thanks.
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This is not a general answer, but it is the answer I ended up using to solve my particular problem. I was interested in knowing if I had an ideal with binomial and monomial generators, if any of the binomials could be replaced by monomials. I just computed the universal Groebner basis, and if my binomial still remained in that basis, it was essential to the first generating set. It seems like this trick should work for more general cases than binomials, though.