Chevalley-Warning theorem

The Chevalley-Warning theorem is a mathematical theorem on solvability of polynomial equations in several variables over a finite field. The theorem was proved by Ewald Warning in 1936. A slightly weaker form of the theorem, known as Chevalley's theorem, was proved by French mathematician Claude Chevalley, also in 1936. Chevalley's theorem settled in the affirmative a conjecture of Artin.

Statement of the theorems

Consider a system of polynomial equations

:$P\_j(x\_1,dots,x\_n)=0,\; j=1,dots,r$

where the $P\_j$ are polynomials with coefficients in a finite field $mathbb\{F\}$ and such that the number of variables satisfies

:$n>sum\_\{j=1\}^r\; d\_j$

where $d\_j$ is the total degree of $P\_j$. The Chevalley-Warning theorem states that the number of common solutions $(a\_1,dots,a\_n)\; in\; mathbb\{F\}^n$ is divisible by the characteristic $p$ of $mathbb\{F\}$. Chevalley's theorem states that if the system has the trivial solution $(0,dots,0)\; in\; mathbb\{F\}^n$, i.e. if the polynomials have no constant terms, then the system also has a non-trivial solution .

Chevalley's theorem is an immediate consequence of the Chevalley-Warning theorem since $p$ is at least 2.

Both theorems are best possible in the sense that, given any $n$, there exists a homogeneous polynomial over $mathbb\{F\}$ of degree $n$ in $n$ variables having only the trivial zero.

Artin's conjecture

It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Austrian mathematician Emil Artin in the early 1930s. The motivation behind Artin's conjecture was the observation, due to Artin, that quasi-algebraically closed fields have trivial Brauer group, in connection with the at that time well-known fact that finite fields have trivial Brauer group (as a consequence of a Wedderburn's theorem).

The Ax-Katz theorem

The Ax-Katz theorem determines more accurately a power $q^b$ of the cardinality $q$ of $mathbb\{F\}$ 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 :$(n\; -\; sum\_j\; d\_j)/d.$

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.

References

*cite journal| last=Chevalley | first=Claude | year=1936 | title= _fr. Démonstration d'une hypothèse de M. Artin | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=11 | pages=73–75 | id=Zbl|0011.14504 JFM|61.1043.01 fr icon
*cite journal| last=Warning | first=Ewald | year=1936 | title= _de. Bemerkung zur vorstehenden Arbeit von Herrn Chevalley | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=11 | pages=76–83 | id=Zbl|0011.14601 JFM|61.1043.02 de icon
*cite journal| last=Ax | first=James | authorlink=James Ax | year=1964 | title=Zeros of polynomials over finite fields | journal=American Journal of Mathematics | volume=86 | pages=255–261 | id=MR|0160775 | doi=10.2307/2373163
*cite book
last = Lang
first = Serge
authorlink =
coauthors = John T. Tate (editors)
title = The collected papers of Emil Artin
publisher = Addison-Wesley
year = 1965
location = Reading, Massachusetts
pages = page x
url =
doi =
id =
isbn =
*cite journal| last=Katz | first=Nicholas M. | authorlink=Nicholas Katz | year=1971 | title=On a theorem of Ax | journal=Amer. J. Math. | volume=93 | issue=2 | pages=485–499 | doi=10.2307/2373389