Ax–Katz theorem

In algebra, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x).

Let ${\displaystyle \mathbb {F} }$ be a finite field and ${\displaystyle \{f_{j}\}_{j=1}^{r}\subseteq \mathbb {F} [X_{1},\ldots ,X_{n}]}$ be a set of polynomials such that the number of variables satisfies

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