= Modular-size auxiliary polynomials
{title2=$q_S(x)=x_S(\sum_i x_i-r)$}
In the <polynomial method in combinatorics>, a <set family> of sets with common size residue $r$ modulo a <prime number> $p$ annihilates the displayed auxiliary functions. After <multilinear reduction on the Boolean cube>, functions with $|S|\leq s-1$ have degree at most $s$. If $r$ avoids every $j=0,\ldots,s-1$ modulo $p$, these functions are <linearly independent>: evaluation at a set $T$ gives $(|T|-r)\sum_{S\subseteq T}c_S$, and increasing $|T|$ eliminates coefficients. Combining them with diagonal <intersection polynomials> removes the lower-degree monomial count and strengthens the dimension bound from $\sum_{j\leq s}\binom nj$ to $\binom ns$.
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