# Minimal generating sets for homogeneous polynomial ideal in two variables

This question is (somehow) related to System of generator 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.

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Denote by $I_m$ the ideal of $A$ generated by $S_m$. Then $I_{m+1}\subset I_m$ and the quotient $I_m/I_{m+1}$ is a vector space of dimension $m+1$ (a basis is given by the classes of the elements of $S_m$). So $I_m$ can't be generated by less than $m+1$ elements.