Modular-size auxiliary polynomials ($q_S(x)=x_S({\sum }_i{x}_i-r)$)

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 $|S|\le s-1$ have degree at most $s$. If $r$ avoids every $j=0,\dots ,s-1$ 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 ${\sum }_{j\le s}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ to $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$.