Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 11 / 3 / Solution

The coefficient form of the Combinatorial Nullstellensatz is as follows. Let $F$ be a field, let $P\in F[x_1,\dots ,x_r]$ have total degree at most $t_1+\cdots +t_r$, and suppose the coefficient of $x_1^{t_1}\cdots x_r^{t_r}$ is nonzero. For finite subsets $S_i\subset F$ with $|S_i|>t_i$, there is a point of $S_1\times \cdots \times S_r$ where $P$ is nonzero.

For completeness, restrict to $|S_i|=t_i+1$. The coefficient identity isBy Lagrange interpolation, the one-variable weighted sum on the right kills powers below $t_i$ and equals one on the power $t_i$. Any monomial of total degree at most $\sum t_i$, other than the specified one, has some exponent below $t_i$ and therefore contributes zero; the specified monomial contributes its coefficient. Every denominator is nonzero in the field. Hence a nonzero coefficient forces a nonzero grid value. The two applications below use this coefficient form.