Combinatorial Nullstellensatz

Sun, 26 Nov 2023 12:27:01 +0000

I came across this fun theorem in a talk about tree colorings:

Combinatorial Nullstellensatz. Let $f \in F[x_1, x_2, \ldots, x_n]$ be a polynomial of degree $t_1 + \cdots + t_n$. If $S_1, S_2, \ldots, S_n$ are nonempty subsets of $F$ such that $\left| S_i \right| \geq t_i + 1$ for all $i$, then there exists $s_1 \in S_1$, $s_2 \in S_2$, $\ldots$, $s_n \in S_n$ for which $$ f(s_1, s_2, \ldots, s_n) \neq 0 $$ as long as the coefficient of $x_1^{t_1} x_2^{t_2} \cdots x_n^{t_n}$ is nonzero.

Problems on PostDoc Problems

Combinatorial Nullstellensatz