Alon-Tarsi lemma
= Alon-Tarsi lemma
{c}
Let $f\in F[x_1,\ldots,x_n]$ have degree at most $d_1+\cdots+d_n$, and let each $A_i\subset F$ contain $d_i+1$ distinct elements. The Alon-Tarsi coefficient formula is
$$
[x_1^{d_1}\cdots x_n^{d_n}]f
=\sum_{a_i\in A_i}
\frac{f(a_1,\ldots,a_n)}
{\prod_i\prod_{b\in A_i\setminus\{a_i\}}(a_i-b)}.
$$
It follows by applying <Lagrange interpolation polynomial>[Lagrange interpolation] successively in each variable.