Let $f \in \mathbb{F}\_q[x\_1, \dots, x\_k]$ be a polynomial with $\deg f = n$, and let $\chi$ be a multiplicative character over $\mathbb{F}\_q$.


Is there any known bound, possibly with conditions about $f$ and $\chi$, for
$$\left|\sum\_{c\_1, \dots, c\_k \in \mathbb{F}\_q} \chi(f(c\_1, \dots, c\_k)) \right| ?$$



Bound for sum of multiplicative character calculated over multivariate polynomial
Yes, there are several known bounds. The following statement, due to Katz, has quite strict conditions on $f$, but gives a very strong result. It is perhaps the simplest statement that gives such a strong bound.


Suppose that the equation $f=0$ defines a nonsingular hypersurface in $\mathbb A^k$, and the degree $n$ part of $f$ defines a nonsingular hypersurface in $\mathbb P^{k-1}$. Suppose also that either $n$ is prime to $q$ or $\chi^n$ is trivial.


Then $$\left|\sum\_{c\_1, \dots, c\_k \in \mathbb{F}\_q} \chi(f(c\_1, \dots, c\_k)) \right| \leq (n-1)^k q^{k/2} $$


This is the main theorem of [Estimates for nonsingular multiplicative character sums](https://doi.org/10.1155/S1073792802106088) by Nick Katz.


