Chevalley–Warning theorem

Let ${\displaystyle \mathbb {F} }$ be a finite field and ${\displaystyle \{f_{j}\}_{j=1}^{r}\subseteq \mathbb {F} [X_{1},\ldots ,X_{n}]}$ be a set of polynomials such that the number of variables satisfies

${\displaystyle n>\sum _{j=1}^{r}d_{j}}$

where ${\displaystyle d_{j}}$ is the total degree of ${\displaystyle f_{j}}$. The theorems are statements about the solutions of the following system of polynomial equations

${\displaystyle f_{j}(x_{1},\dots ,x_{n})=0\quad {\text{for}}\,j=1,\ldots ,r.}$

• Chevalley–Warning theorem states that the number of common solutions ${\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}}$ is divisible by the characteristic ${\displaystyle p}$ of ${\displaystyle \mathbb {F} }$. Or in other words, the cardinality of the vanishing set of ${\displaystyle \{f_{j}\}_{j=1}^{r}}$ is ${\displaystyle 0}$ modulo ${\displaystyle p}$.
• Chevalley's theorem states that if the system has the trivial solution ${\displaystyle (0,\dots ,0)\in \mathbb {F} ^{n}}$, i.e. if the polynomials have no constant terms, then the system also has a non-trivial solution ${\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}\backslash \{(0,\dots ,0)\}}$.

Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since ${\displaystyle p}$ is at least 2.

Both theorems are best possible in the sense that, given any ${\displaystyle n}$, the list ${\displaystyle f_{j}=x_{j},j=1,\dots ,n}$ has total degree ${\displaystyle n}$ and only the trivial solution. Alternatively, using just one polynomial, we can take f₁ 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.

Warning proved another theorem, known as Warning's second theorem, which states that if the system of polynomial equations has the trivial solution, then it has at least ${\displaystyle q^{n-d}}$ solutions where ${\displaystyle q}$ is the size of the finite field and ${\displaystyle d:=d_{1}+\dots +d_{r}}$. Chevalley's theorem also follows directly from this.

Remark: If ${\displaystyle i<q-1}$ then

${\displaystyle \sum _{x\in \mathbb {F} }x^{i}=0}$

so the sum over ${\displaystyle \mathbb {F} ^{n}}$ of any polynomial in ${\displaystyle x_{1},\ldots ,x_{n}}$ of degree less than ${\displaystyle n(q-1)}$ also vanishes.

The total number of common solutions modulo ${\displaystyle p}$ of ${\displaystyle f_{1},\ldots ,f_{r}=0}$ is equal to

${\displaystyle \sum _{x\in \mathbb {F} ^{n}}(1-f_{1}^{q-1}(x))\cdot \ldots \cdot (1-f_{r}^{q-1}(x))}$

because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials ${\displaystyle f_{i}}$ is less than n then this vanishes by the remark above.

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, named after James Ax and Nicholas Katz, determines more accurately a power ${\displaystyle q^{b}}$ of the cardinality ${\displaystyle q}$ of ${\displaystyle \mathbb {F} }$ dividing the number of solutions; here, if ${\displaystyle d}$ is the largest of the ${\displaystyle d_{j}}$, then the exponent ${\displaystyle b}$ can be taken as the ceiling function of

${\displaystyle {\frac {n-\sum _{j}d_{j}}{d}}.}$

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 ${\displaystyle q}$ divides each of these algebraic integers.

• Combinatorial Nullstellensatz

• Artin, Emil (1982), Lang, Serge.; Tate, John (eds.), Collected papers, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90686-7, MR 0671416
• Ax, James (1964), "Zeros of polynomials over finite fields", American Journal of Mathematics, 86: 255–261, doi:10.2307/2373163, MR 0160775
• Chevalley, Claude (1935), "Démonstration d'une hypothèse de M. Artin", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in French), 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 (1935), "Bemerkung zur vorstehenden Arbeit von Herrn Chevalley", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German), 11: 76–83, doi:10.1007/BF02940715, JFM 61.1043.02, Zbl 0011.14601
• Serre, Jean-Pierre (1973), A course in arithmetic, pp. 5–6, ISBN 0-387-90040-3

• "Proof's of the Chevalley-Warning theorem".