= Homogenisation on a uniform layer
= Homogenization on a uniform layer
{synonym}
On the <uniform layer of the Boolean cube> $\Omega_r$, every <multilinear polynomial> of <polynomial degree> at most $s\leq r$ is in the <span> of the degree-$s$ square-free <monomials>. For $T\subseteq[n]$, $|T|=j\leq s$, and $x_T=\prod_{i\in T}x_i$, the identity is
$$
x_T=\binom{r-j}{s-j}^{-1}\sum_{\substack{S\supseteq T\\|S|=s}}x_S\quad\text{on }\Omega_r.
$$
At the <characteristic vector of a set> $A$ of size $r$, both sides vanish if $T\nsubseteq A$. Otherwise exactly $\binom{r-j}{s-j}$ summands are $1$. Hence the <dimension> of the restricted polynomial space is at most $\binom ns$. No assertion of independence of the spanning <monomials> is needed, so no condition $r+s\leq n$ is required. Unlike ordinary <homogenization>, this operation preserves degree bounds by using the fixed-weight evaluation domain.
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