Chevalley–Warning theorem

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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[edit]

Let \mathbb{F} be a finite field and \{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

n>\sum_{j=1}^r d_j

where d_j is the total degree of f_j. The theorems are statements about the solutions of the following system of polynomial equations

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 (a_1,\dots,a_n) \in \mathbb{F}^n is divisible by the characteristic p of \mathbb{F}. Or in other words, the cardinality of the vanishing set of \{f_j\}_{j=1}^r is 0 modulo p.
  • 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 f_j = x_j, j=1,\dots,n has total degree 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 + ... + xnan where the elements a form a basis of the finite field of order pn.

Proof of Warning's theorem[edit]

Let q=|\mathbb{F}|.

Remark: If i<q-1 then

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

so the sum over \mathbb{F}^n of any polynomial in x_1,\ldots,x_n of degree less than n(q-1) also vanishes.

The total number of common solutions modulo p of f_1, \ldots, f_r = 0 is equal to

\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 f_i is less than n then this vanishes by the remark above.

Artin's conjecture[edit]

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[edit]

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.

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