Let $\mathbb{F}\_q$ be a finite field, $\psi$ be a non-trivial additive character over $\mathbb{F}\_q$, and $a, b \in \mathbb{F}\_q$ constants. Is there any known estimate for the gaussian sum


$$\sum\_{x \in \mathbb{F}\_q} \psi( a x^m + b x^n),$$


possibly for specific values of $m, n \in \mathbb{Z}\_{\ge 2}$, $m \ne n$?



Known estimate for gaussian sum $\sum_{x \in \mathbb{F}_q} \psi( a x^m + b x^n)$?
As Ofir Gorodetsky notes in the [comments](https://mathoverflow.net/questions/422709/known-estimate-for-gaussian-sum-sum-x-in-mathbbf-q-psi-a-xm-b-xn#comment1086256_422709), Weil's bound gives that the absolute value of the sum is most order $\max(m,n) \sqrt{q}$. This is non-trivial as long as $m$ and $n$ are $o(\sqrt{q})$, after this the estimate is worse than the trivial bound.


When $m$ and $n$ are large there are known estimates from additive combinatorics, at least when $q$ is prime. There are some obstructions to estimates, particularly when $m$, $n$ or $m-n$ have a large common factor with $q-1$. Otherwise one can obtain cancellation. See: J. Bourgain's paper "[Mordell's Exponential Sum Estimate Revisited](https://doi.org/10.1090/S0894-0347-05-00476-5)".


