= Solution
The <coefficient> form of the <Combinatorial Nullstellensatz> is as follows. Let $F$ be a <field>, let $P\in F[x_1,\ldots,x_r]$ 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 is
$$
[x_1^{t_1}\cdots x_r^{t_r}]P
=\sum_{a_i\in S_i}\frac{P(a_1,\ldots,a_r)}{\prod_i\prod_{b\in S_i\setminus\{a_i\}}(a_i-b)}.
$$
By <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.
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