Proofs of the Chevalley-Warning Theorem

I am working on a book-length manusript, Around the Chevalley-Warning Theorem. A complete answer to your question is estimated at about 150 pages!

In terms of what exists at the moment, here are two papers. Both of them make connections between the classical results of Chevalley and Warning and modern polynomial methods. The first concerns a generalization of the (unjustly almost forgotten) Warning's Second Theorem to restricted variables. The second explains the connection between Chevalley's Theorem and Alon's Combinatorial Nullstellensatz. I take the perspective that the Combinatorial Nullstellensatz is in fact a very direct generalization of Chevalley's proof of Chevalley's Theorem. (I don't mean "very direct" as a slight against Alon: I am certainly a fan. Rather it is meant to indicate a useful -- at least for a number theorist -- way of thinking about these results.)

I'm afraid the above seems overly self-promotional. Let me also give what I think are the most important papers in this area, with an emphasis on relatively elementary work. [So I will not list e.g. work of Esnault, though I agree with Daniel Loughran's suggestion that it is, at least in some sense, the most important result of Chevalley-Warning type.]

J. Ax, \emph{Zeroes of polynomials over finite fields}. Amer. J. Math. 86 (1964), 255-–261.

C. Chevalley, \emph{D\'emonstration d’une hypoth`ese de M. Artin.} Abh. Math. Sem. Univ. Hamburg 11 (1935), 73–-75.

D.R. Heath-Brown, \emph{On Chevalley-Warning theorems}. (Russian. Russian summary) Uspekhi Mat. Nauk 66 (2011), no. 2(398), 223--232; translation in Russian Math. Surveys 66 (2011), no. 2, 427-–436.

D.J. Katz, \emph{Point count divisibility for algebraic sets over $\mathbb{Z}/p^{\ell}\mathbb{Z}$ and other finite principal rings}. Proc. Amer. Math. Soc. 137 (2009), 4065-–4075.

N.M. Katz, \emph{On a theorem of Ax}. Amer. J. Math. 93 (1971), 485-–499.

M. Marshall and G. Ramage, \emph{Zeros of polynomials over finite principal ideal rings}. Proc. Amer. Math. Soc. 49 (1975), 35-–38.

O. Moreno and C.J. Moreno, \emph{Improvements of the Chevalley-Warning and the Ax-Katz theorems}. Amer. J. Math. 117 (1995), 241--244.

S.H. Schanuel, \emph{An extension of Chevalley's theorem to congruences modulo prime powers}. J. Number Theory 6 (1974), 284-–290.

G. Terjanian, \emph{Sur les corps finis}. C. R. Acad. Sci. Paris S\'er. A-B 262 (1966), A167-–A169.

D.Q. Wan, \emph{An elementary proof of a theorem of Katz}. Amer. J. Math. 111 (1989), 1-–8.

E. Warning, \emph{Bemerkung zur vorstehenden Arbeit von Herrn Chevalley}. Abh. Math. Sem. Hamburg 11 (1935), 76–-83.

Less than a year ago I thought there were about ten papers which generalize and refine the Chevalley-Warning Theorem. I now think I was off by a full order of magnitude, and indeed my current bibliography contains about 100 references. (I will admit that at this point, the radius of the circle referred to in Around the Chevalley-Warning Theorem is rather large. It includes for instance material on Davenport constants and on polynomial interpolation.)

Added: To answer the question more directly: Chevalley's proof critically uses the observation that if $P_1,\ldots,P_r$ are polynomials in $n$ variables over $\mathbb{F}_q$, then the function

$x \in \mathbb{F}_q^n \mapsto \chi(x_1,\ldots,x_n) = \prod_{i=1}^r(1-P_i(x_1,\ldots,x_n)^{q-1})$

is the characteristic function of the set

$Z = \{ (x_1,\ldots,x_n) \in \mathbb{F}_q^n \mid P_1(x_1,\ldots,x_n) = \ldots = P_r(x_1,\ldots,x_n) = 0\}$.

Every other proof of Chevalley's Theorem I know uses this observation. The subsequent proofs of Chevalley's Theorem other than Ax's proof look (to me) essentially the same as Chevalley's. Ax's proof uses (only!) Chevalley's observation and Ax's Lemma: if $\operatorname{deg} P < (q-1)n$, then $\sum_{x \in \mathbb{F}_q^n} P(x) = 0$. Ax's Lemma is impressively easy to prove: it would be a fair question on many undergraduate algebra midterms. I think I saw somewhere the claim that it goes back to V. Lebesgue. I still cannot quite see Ax's argument as a recasting of Chevalley's. So after all this I suppose I would say that there are "really two proofs".

One of the best results in this vein which I know of is due to Hélène Esnault, and appears in the paper:

Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. math. 151 (2003), 187-191.

One particularly nice result in this paper is Corollary 1.3, which states that any Fano variety over a finite field, or more generally any chain rationally connected variety, admits a rational point. The proof naturally uses the full force of étale cohomology.

It is not quite clear to me that this recovers the original Chevalley–Warning theorem in its full generality, as she assumes that her varieties are smooth and projective. But this certainly recovers the special case of the Chevalley–Warning theorem when the variety in question is smooth and projective.

Regarding the "weak" form of Chevalley-Warning - namely that the number of solutions is not exactly one, rather than being necessarily divisible by the characteristic - there is a combinatorial proof using Alon's "combinatorial nullstellensatz", as was already noted in Alon's original paper. But the nullstellensatz is significantly more general than Chevalley-Warning and has a large number of applications in combinatorics.