When is Chevalley Warning's bound best possible? Chevalley Warning's theorem (a form of) states that any homogeneous form over a finite field of degree $d$ in more than $d$ variables has a nontrivial zero in the field. However, for diagonal forms, this result is often much weaker than it needs to be. I am wondering if anyone knows of a particularly good reference on improving the bound for certain families of diagonal forms? I apologize if this is a question better fit for MSE.
 A: Consider a degree $d$ homogeeous diagonal form in $n$ variables over $\mathbb F_q$.
I think that for $d$ quite large this is a problem in additive combinatorics. Essentially you have sets $A_1, \dots, A_n$, each a translate of the set of $d$th powers, and you're trying to show that $0 \in A_1+ \dots + A_n$ nontrivially.
One possible method is Fourier analytic. Without loss of generality $d$ divides $q-1$.  We may write the number of solutions of $\sum_{i=1}^n a_i x_i^{d}=0$ as
$$\frac{1}{q} \sum_{\psi: \mathbb F_q \to \mathbb C^{\times}}\prod_{i=1}^n \left( \sum_{x_i \in \mathbb F_q }\psi( a_i x_i^d) \right)$$
The main term $\psi=1$ is $q^{n-1}$, so to have the other terms smaller and thus guarantee nontrivial solutions we need an estimate for $\psi$ nontrivial:
$$ \left|\sum_{x \in \mathbb F_q }\psi( a_i x_i^d)\right| \leq q^{1-1/n}$$
The Weil bound gives the estimate:
$$\left|\sum_{x \in \mathbb F_q }\psi( a_i x_i^d)\right| \leq (d-1) \sqrt{q}$$
so for $d  \leq q^{1/2-1/n}$ we win.
This can be improved when $q$ is prime using better estimates for character sums, e.g. Estimates for the number of sums and products and for exponential sums in fields of prime order by Bourgain, Gibichuk, and Konyagin.
However I suspect that the Gauss sum method is not optimal. For $q$ a perfect square and $d= \sqrt{q}+1$, say, there is no cancellation in the Gauss sums, but still for $n=3$ we get a nontrivial solution by linear algebra over $\mathbb F_{q^{1/2}}$.
