= Solution
The <Alon-Tarsi lemma> says that if $\deg f\leq d_1+\cdots+d_n$ and $A_i$ consists of $d_i+1$ distinct field elements, then
$$
[x_1^{d_1}\cdots x_n^{d_n}]f
=\sum_{a_i\in A_i}
\frac{f(a_1,\ldots,a_n)}
{\prod_i\prod_{b\in A_i\setminus\{a_i\}}(a_i-b)}.
$$
This follows by applying univariate Lagrange interpolation successively in each variable.
If the displayed coefficient is nonzero, at least one summand has $f(a_1,\ldots,a_n)\ne0$. This is the coefficient form of the <Combinatorial Nullstellensatz>.
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