Polynomial representation of a graph

Choose points $a_v\in F^r$ and polynomials $f_v$ in a finite-dimensional polynomial space $M$, with $f_v(a_v)\ne 0$ and $f_v(a_w)=0$ for distinct nonadjacent vertices. Evaluation produces a fitting matrix of rank at most ${\dim }_{F}M$. Equivalently, tensoring the polynomial spaces gives linearly independent polynomials for every independent code in a strong power, proving $c(G)\le {\dim }_{F}M$.