quadratic character sum over a subfield

Question

Let $q$ be an odd prime power and let $\chi$ be the quadratic character on $\mathbb{F}_{q^2}$. Let $f(x)$ be a univariate polynomial with coefficients in $\mathbb{F}_{q^2}$. I am interested in any Weil-type bounds for the character sum $\sum_{x\in\mathbb{F}_q} \chi(f(x))$. The summation is over the subfield $\mathbb{F}_q$ only. Thank you for any references that you may have.

Answer 1

There is a Weil-type bound for such sums, but I am uncertain whether it has been published. I once needed it in a special case (my $f$ was actually a rational function, but that is not really relevant). I had worked hard to prove the case I needed by studying the relevant $L$-functions. After describing my calculations at a conference, W.C. Li, Penn State came forward and described a more general variant. At the same conference Robert Odoni, University of Glasgow also got in touch with me, and told that he, too, is aware of the same result, and was working on generalizations with his grad students. I am not in a position to judge who did what first, and whether anything has been published (the incident took a lot of wind out of my sails but let's not go there). Anyway here's my recollection of the result (not 100% about all the details).

Let $\overline{f}$ be the $\Bbb{F}_q$-conjugate of $f$. Let $m$ be the number of distinct poles and zeros of $f \overline{f}$. Then, for any multiplicative character $\chi$ of $\Bbb{F}_{q^2}^*$ we have the bound $$ \left|\sum_{x\in\Bbb{F}_q}\chi(f(x))\right|\le N \sqrt q, $$ where $N=m-1$ if infinity is either a zero or a pole of $f$, and $N=m-2$ if infinity is neither zero nor a pole of $f$ (if $f$ is a polynomial, then infinity is automatically a pole).

Caveats:

• The above description is probably missing a few key assumptions. For example, we need to exclude trivial sums. Say if $f$ is a square and $\chi$ is of order two, then obviously the sum is trivial. So the usual assumptions about cancelling any eventual factor of $f$ of multiplicity divisible by the order of $\chi$ certainly applies.
• I am a bit ashamed to tell that I don't know if either of them published the result anywhere. I was left with the impression that the result is relatively straightforward to anyone well versed in the class field theory of function fields (which would explain the reluctance of senior people to publish it). If needing this desperately I might dig up the dissertations of Odoni's graduate students from the 90s. May be one of them used and described this result?