Chevalley-Warning Theorem

Let ${𝔽}_q$ be the finite field of $q$ elements withcharacteristic $p$. Let $f_i(x_1,\mathrm{\dots },x_n)$, $i=1,2,\mathrm{\dots },r$,be polynomial of $n$ variables over ${𝔽}_q$. If $n>{\sum }_{i=1}^r\mathrm{deg}(f_i)$, then the number of solutions over${𝔽}_q$ to the system of equations

$f_1(x_1,x_2,\mathrm{\dots },x_n)$$=0$

$f_2(x_1,x_2,\mathrm{\dots },x_n)$$=0$

$\mathrm{⋮}$

$f_r(x_1,x_2,\mathrm{\dots },x_n)$$=0$

is divisible by $p$. In particular, if none of the polynomials$f_1$, $f_2,\mathrm{\dots },f_r$ have constant term, then there are at least$p$ solutions.