Polynomial method in combinatorics

The polynomial method in combinatorics encodes a discrete configuration by polynomials and obtains combinatorial bounds from their degree, zeros, coefficients, or linear independence.

Let $f\in F[x_1,\dots ,x_n]$ 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 isIt follows by applying Lagrange interpolation successively in each variable.

One coefficient form of the Combinatorial Nullstellensatz says that if $\mathrm{deg}f\le d_1+\cdots +d_n$ and $[x_1^{d_1}\cdots x_n^{d_n}]f\ne 0$, then for every choice of sets $A_i$ with $|A_i|=d_i+1$ there is a point $(a_1,\dots ,a_n)\in A_1\times \cdots \times A_n$ at which $f$ does not vanish.