Chevalley–Warning theorem

Statement

Let $\mathbb{F}$ be a finite field of characteristic $p$. Let $f_1, \dots, f_r \in \mathbb{F}[X_1, \dots, X_n]$ be polynomials with total degrees $d_1, \dots, d_r$, such that $n > d_1 + \dots + d_r$. Then the number of common zeros $(a_1, \dots, a_n) \in \mathbb{F}^n$ of $f_1, \dots, f_r$ is divisible by $p$.

Remarks

A corollary is Chevalley's theorem: if the polynomials have no constant terms, then they have a common zero other than $(0, \dots, 0)$. The reason is that $(0, \dots, 0)$ is a zero, and the number of zeros is a multiple of $p \ge 2$. For example, a quadratic form in $3$ or more variables over $\mathbb{Z}/p\mathbb{Z}$ has a nontrivial zero.

History

Ewald Warning and Claude Chevalley proved the results in 1935.

Explanations

L1 human

When a system of polynomial equations over a finite field has many variables compared with the degrees, the solutions come in multiples of $p$. So if there is one obvious solution, such as all zeros, there must be another.