Chevalley-Warning theorem

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In mathematics, Chevalley's theorem on solutions of polynomial equations over a finite field $\mathbb{F}$ with $q$ elements, $q$ a power of the prime number $p$ , states that for a polynomial

$a(x_1, \ldots, x_n)$

of total degree $d$ , with

d < n

the number of solutions to

$a(x_1, \ldots, x_n) = 0$

in $\mathbb{F}^n$ is divisible by $p$ . As a corollary, if $a$ is a homogeneous polynomial of degree $d$ , then there is at least one other solution than

$(0, \ldots,0)$ .

This was proved in 1936 by Claude Chevalley in response to a question of Emil Artin. There is an elementary proof using the $p$ -th power map.

Warning's generalisation is for several polynomials $a j$ and simultaneous equations, under the condition that the sum of the total degrees $d j$ is less than $n$ . It equally has the corollary that when the polynomials are homogeneous, there is a non-zero solution.

The Ax-Katz theorem determines more accurately a power $q b$ dividing the number of solutions; here, if $d$ is the largest of the $d j$ , then the exponent $b$ can be taken as the ceiling function of

(n −	∑	d_j) / d
	j

The Ax-Katz result has an interpretation in étale cohomology as a divisibility result for the (reciprocals of) the zeroes and poles of the local zeta-function. Namely, the same power of $q$ divides each of these algebraic integers.

[edit] See also

Quasi-algebraic closure

[edit] References

Chevalley, Claude (1936). "French: Démonstration d'une hypothèse de M. Artin". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11: 73–75. Zbl 0011.14504 JFM 61.1043.01. (French)
Warning, Ewald (1936). "Comment on the preceding work of Mr. Chevalley (German: Bemerkung zur vorstehenden Arbeit von Herrn Chevalley)". Abhand. Mathe. Sem. Hamburg 11: 76–83. Zbl 0011.14601 JFM 61.1043.02. (German)
Ax, James (1964). "Zeros of polynomials over finite fields". American Journal of Mathematics 86: 255–261. doi:10.2307/2373163. MR 0160775.
Katz, Nicholas M. (April 1971). "On a theorem of Ax". Amer. J. Math. 93 (2): 485–499. doi:10.2307/2373389.