Chevalley–Warning 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 (1936) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1936). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x).

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 $(a_1,\dots,a_n) \in \mathbb{F}^n \backslash \{(0,\dots,0)\}$.

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, the list $P_j = x_j, j=1,\dots,n$ has total degree n and only the trivial solution. Alternatively, using just one polynomial, we can take P₁ to be the degree n polynomial given by the norm of x₁a₁ + ... + x_na_n where the elements a form a basis of the finite field of order pⁿ.

Proof of Warning's theorem

If i<p−1 then

$\sum_{x\in\mathbb{F}}x^i=0$

so the sum over Fⁿ of any polynomial in x₁,...,x_n of degree less than n(p−1) also vanishes.

The total number of common solutions mod p of P₁ = ... = P_r = 0 is

$\sum_{(x_1,\cdots,x_n)\in\mathbb{F}^n}(1-P_1^{p-1})\times\cdots\times(1-P_r^{p-1})$

because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials P_i is less than n then this vanishes by the remark above.

Artin's conjecture

It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn's theorem.

The Ax–Katz theorem

The Ax–Katz theorem, named after James Ax and Nicholas Katz, 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

$\frac{n - \sum_j d_j}{d}.$

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

• Artin, Emil (1982), Lang, Serge.; Tate, John, eds., Collected papers, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90686-7, MR671416 
• Ax, James (1964), "Zeros of polynomials over finite fields", American Journal of Mathematics 86: 255–261, doi:10.2307/2373163, MR0160775 
• Chevalley, Claude (1936), "Démonstration d'une hypothèse de M. Artin" (in French), Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11: 73–75, doi:10.1007/BF02940714, JFM 61.1043.01, Zbl 0011.14504 
• Katz, Nicholas M. (1971), "On a theorem of Ax", Amer. J. Math. 93 (2): 485–499, doi:10.2307/2373389 
• Warning, Ewald (1936), "Bemerkung zur vorstehenden Arbeit von Herrn Chevalley" (in German), Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11: 76–83, doi:10.1007/BF02940715, JFM 61.1043.02, Zbl 0011.14601