This question is (somehow) related to System of generators of a homogenous ideal
Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let
$ {\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, \ldots ,Y^m \rbrace $
and let $I$ be the ideal of $A$ generated by ${\cal S}_m$. Now let $S'$ be another finite set of polynomials in $A$ such that $I$ is also the ideal of $A$ generated by $S'$. Does it necessarily follow that $S'$ has at least $m+1$ elements ? Note that $I$ is an homogeneous ideal, and perhaps the theory of Groebner bases can help here, though I don't see how.