TITLE: Is there a proof of Warning's Second Theorem using p-adic cohomology?
QUESTION [13 upvotes]: Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$.  Put $d = \sum_{i=1}^n d_i$ and suppose $d< n$. Let $V(f_1,\ldots,f_r) = \{ x \in \mathbb{F}_q^n \mid f_1(x) = \ldots = f_r(x) = 0\}$.  
The papers

C. Chevalley, Demonstration d’une hypothese de M. Artin Abh. Math. Sem. Univ. Hamburg 11 (1935), 73–-75

and 

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

contain the following results:
Chevalley's Theorem: We have $\# V(f_1,\ldots,f_r) \neq 1$.  
Warning's First Theorem: We have $\operatorname{char}(F) \mid \# V(f_1,\ldots,f_r)$.  
Warning's Second Theorem: If $V(f_1,\ldots,f_r)$ is nonempty, its cardinality is at least $q^{n-d}$.
The first two results are often combined as the "Chevalley-Warning Theorem", and the last result seems to be too often overlooked.  The Chevalley-Warning Theorem is the beginning of a long story of p-adic estimates on the number of $\mathbb{F}_q$-rational points on such $\mathbb{F}_q$-schemes, culminating in the Ax-Katz Theorem which determines the minimal $p$-adic valuation of $\# V(f_1,\ldots,f_r)$ as $f_1,\ldots,f_r$ range over all polynomials of degrees $d_1,\ldots,d_r$.  Famously, Katz used p-adic cohomology to prove his theorem.  Other, more elementary proofs have since been given, but this approach remains a very natural one.

Question: Is there a proof of Warning's Second Theorem using $p$-adic cohomology?  Does such a proof appear in the literature?  

Warning's proof of his second theorem is an ingenious elementary argument of the sort that nowadays goes under the moniker "Polynomial Method".  See this paper of Heath-Brown which includes Warning's proof as a point of departure to further work.  (To the best of my knowledge, Heath-Brown's paper is the only one which directly follows up on Warning's Second Theorem.  I would be very interested to learn of others.)
A more modern Polynomial Method approach to Warning's Second Theorem which leads to a different kind of generalization is discussed here.
Added: To address Daniel Litt's question raised in the comments: I do not know how to deduce Warning's Second Theorem (or even his first, for that matter) from the Weil Conjectures.  The fact that we are making no "geometric" assumptions on our affine subscheme would make such a deduction a forbidding endeavor for me....but I do not claim that it cannot be done.  I would be happy to broaden the question to ask about a cohomological proof of Warning II: you get to pick the cohomology theory.

REPLY [5 votes]: I do not think that there is such a proof in the litterature, but as you have guessed, $p$-adic cohomology has something to do with it.
Let $X=V(f_1,\dots,f_r)$. Let $K$ be the field of fractions of the Witt vectors of $k$, and let $\phi$ be the unique automorphism of $K$ such that $\phi(a)\equiv a^p \pmod p$. The rigid cohomology groups with compact supports of $X$, $H^i_{\rm rig,c}(X/K)$, are 
$\phi$-isocrystals, namely finite dimensional $K$-vector spaces  endowed with a $\phi$-linear automorphism $\Phi$.
If $k=\mathbf F_q$, with $q=p^r$, the Lefshetz trace formula in rigid cohomology asserts that 
$$ \# X(k)  = \sum (-1)^i \mathop{\rm Tr}(\Phi^r|H^i_{\rm rig,c}(X/K)). $$
(Note that $\Phi^r$ is linear, so that the trace makes sense.)
The theory of slopes of $\Phi$-isocrystals shows that Warning's theorem would follow from the condition that all slopes of $H^i_{\rm rig,c}(X/K)$ 
are $\geq 1$. As you said, it does not seem to be known without any smoothness assumption. Moreover, there might be cancellations in the trace formula that make this expectation wrong (see however this paper of Esnault and Katz).
However, there is a nice theorem of Berthelot, Esnault and Rülling in this direction: Let $X$ be a flat, regular and proper $R$-scheme whose generic fiber $X_K$ is geometrically connected and satisfies $H^i(X_K,\mathscr O_{X_K})=0$ for $i>0$. Then $\# X(k)\equiv 1\pmod q$.  The hard part of the proof consists in proving that the Witt vector cohomology vanishes in positive degrees.