# Chevalley-Warning Theorem

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements with characteristic $p$. Let $f_{i}(x_{1},\ldots,x_{n})$, $i=1,2,\ldots,r$, be polynomial of $n$ variables over $\mathbb{F}_{q}$. If $n>\sum_{i=1}^{r}\deg(f_{i})$, then the number of solutions over $\mathbb{F}_{q}$ to the system of equations

 $\displaystyle f_{1}(x_{1},x_{2},\ldots,x_{n})$ $\displaystyle=0$ $\displaystyle f_{2}(x_{1},x_{2},\ldots,x_{n})$ $\displaystyle=0$ $\displaystyle\vdots$ $\displaystyle f_{r}(x_{1},x_{2},\ldots,x_{n})$ $\displaystyle=0$

is divisible by $p$. In particular, if none of the polynomials $f_{1}$, $f_{2},\ldots,f_{r}$ have constant term, then there are at least $p$ solutions.

Title Chevalley-Warning Theorem ChevalleyWarningTheorem 2013-03-22 17:46:52 2013-03-22 17:46:52 kshum (5987) kshum (5987) 6 kshum (5987) Theorem msc 12E20