Modular intersection polynomial ($P_A(x)={\prod }_{\lambda \in L}({\sum }_{i\in A}{x}_i-\lambda )$)

If member sizes modulo a prime number $p$ avoid $L$ while distinct intersections lie in $L$, evaluation at characteristic vectors of sets gives a diagonal matrix with nonzero diagonal. After multilinear reduction on the Boolean cube, these polynomials remain linearly independent and have degree at most $|L|$. Counting square-free monomials proves $|\mathcal{F}|\le {\sum }_{j=0}^{|L|}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$. On a fixed uniform layer, the low-degree evaluation rank on a uniform layer sharpens this to $\left(\genfrac{}{}{0ex}{}{n}{|L|}\right)$ when $|L|$ does not exceed the layer rank.