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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 $$\left|\sum_{c_1, \dots, c_k \in \mathbb{F}_q} \chi(f(c_1, \dots, c_k)) \right| ?$$

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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 by Nick Katz.

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  • $\begingroup$ Is there a keyword (I would have guessed multivariate Weil sums, but that doesn't seem to be in use) that is useful for looking up the literature on sums of this type? $\endgroup$
    – Anurag Sahay
    Commented Apr 10, 2022 at 2:58
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    $\begingroup$ @AnuragSahay I don't know one other than "exponential sums". Another paper to look at is Estimates for Singular Multiplicative Character Sums by Antonio Rojas-León. doi.org/10.1155/IMRN.2005.1221 $\endgroup$
    – Will Sawin
    Commented Apr 10, 2022 at 12:56
  • $\begingroup$ thanks for the reference! $\endgroup$
    – Anurag Sahay
    Commented Apr 10, 2022 at 13:44

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