For a fixed degree $d>1$, let $\{x^\alpha: |\alpha|=d\}$ be the set of monomials of degree $d$ in the variables $x_1,\ldots, x_n$. View these monomials as $N=\binom{n+d-1}{d-1}>n$ complex polynomials in ${\mathbb{C}}[x_1,\ldots, x_n]$. For each collection $I$ of multi-indices of length $d$, consider the subscheme $\bigcap_{\alpha\in I}\{x^\alpha=0\}$ in affine space $\mathbb{C}^N$. As we let $I$ range over collections of at most $n$ multi-indices, these subschemes are nonempty and distinct from each other, i.e. these intersections have distinct zero sets considering multiplicity.
I'm wondering if we can extend this property to the entire collection. Is it possible to find $N$ polynomials $g_1(x_1,\ldots, x_N),\ldots, g_N(x_1,\ldots, x_N)$ such that
(1) $g_\alpha(x_1,\ldots,x_n,0,\ldots, 0)=x^\alpha$. (i.e. $g_i$ is formed from $x^\alpha$ by adding extra variables)
(2) As we range over arbitrary $I$, the subschemes $\bigcap_{\alpha\in I}\{g_\alpha=0\}$ are non-empty and distinct from each other?
