 Polynomial evaluation map on a finite point set

Evaluation of bounded-degree multivariate polynomials on a finite set defines a linear map $E_P:f\mapsto (f(p){)}_{p\in P}$. Its rank is at most $|P|$ and its kernel is the space of polynomials vanishing on $P$. The rank-nullity theorem gives $\dim \ker E_P\ge \left(\genfrac{}{}{0ex}{}{d+n}{n}\right)-|P|$ for degree at most $d$ in $n$ variables. Dependent point constraints only enlarge the kernel.