Is there an ideal decomposition that counts the number of monomial generators?

Question

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.