Let $S=k[x_1,...,x_n]$, where $k$ is an algebraically closed field of characteristic $0$. Let $\mathfrak m=(x_1,...,x_n)$ be the unique homogeneous maximal ideal of $S$. Then $\mu(\mathfrak m^2)=\binom{n+1}{2} $.
If $J$ is an ideal of $S$, minimally generated by a set $A$ consisting of $t$ many degree $2$ homogeneous polynomials, then is it true that $A$ can be extended to a minimal homogeneous generating set for $\mathfrak m^2$ (so in particular, is it true that $t \le \binom{n+1}{2} $) ?
My claim is equivalent to the following claim : Let $\mathfrak n$ be the image of $\mathfrak m$ in $R:=S/J$ where $J$ is minimally generated by $t$ many degree $2$ homogeneous polynomials. Then $\mu(\mathfrak n^2)=\mu(\mathfrak m^2)-t $.
I can prove this if $J$ is minimally generated by $t$ many degree $2$ monomials, but otherwise I'm not sure.
Please help.
I assume $\mu(J)$ is the minimal number of generators of $J$.
Suppose $J$ is an ideal generated by degree-2 homogeneous polynomials. Given a minimal set $A$ of such generators, it should be easy to show that $\{f + \mathfrak{m}^3 : f \in A\}$ is a basis of $(J+\mathfrak{m}^3)/\mathfrak{m}^3$ (as a $k$-vector space). Conversely, any basis of $(J+\mathfrak{m}^3)/\mathfrak{m}^3$ gives a minimal generating set of $J$ by picking the unique representatives of minimum degree (and these are inverse constructions). Now any basis of $(J+\mathfrak{m}^3)/\mathfrak{m}^3$ extends to a basis of $\mathfrak{m}^2/\mathfrak{m}^3$, which should give the result.