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Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define $$ K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)), $$ where $e_p(t):=e^{2\pi i \frac{t}{p}}$.

It is well-known that if $p\nmid d$, then $|K|$ is bounded by $1$ (up to some constant depending on $d$).

Now consider a sum of products for fixed $t\neq 0$ and $y\neq y'$: $$ I=\sum_{x\in\mathbb{F}_p}K(x,y)\overline{K(x-t,y+t)}\overline{K(x,y')}K(x-t,y'+t). $$ My question is: for what $f\in\mathbb{F}_p[X]$ can we bound $|I|$ by $\sqrt{p}$ (up to a multiplicative constant, of course), or perhaps a little worse: $p^{\delta}$ ($\delta<1$). When $f$ is quadratic, then $K$ is a Gauss sum that can be evaluated explicitly, and I can calculate $I$ and answer my question affirmatively. I don't know what happens for general $f$.

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Here is a different approach than Denis Chaperon de Lauzières's.

Opening everything and using orthogonality of characters to remove the $x$ variable, we see that your sum is $1/p$ times $$\sum_{z_1,z_2,z_3,z_4 \in \mathbb F_p, z_1-z_2-z_3+z_4 =0 } e_p \left( y f(z_1) + t z_2 - (y+t) f(z_2) - y' f(z_3) -t z_4 + (y'+t) f(z_4) \right)$$

This is an exponential sum of a $3$-variable polynomial. Deligne's theorem (Theorem 8.4 of Weil I) shows that this is $O(p^{3/2})$ (with explicit constant $(d-1)^{3}$) as long as $d$ is prime to $p$ and the leading term of the polynomial inside the $e_p$ defines a smooth hypersurface.

The leading term looks like $$y z_1^d - (y+t) z_2^d - y' z_3^d + (y'+ t) z_4 ^d $$

The intersection of this with $z_1 - z_2 - z_3 + z_4 =0$ is smooth unless $$ y^{-1/(d-1)} - (y+t)^{-1/(d-1)} - y' ^{-1/(d-1)} + (y'+t)^{1/d-1}=0$$.

So you get square-root cancellation outside this set for arbitrary $f$.

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  • $\begingroup$ Thank you for the detailed calculation. For each $y$, there are only $O(1)$ $y'$ that make the hypersurface non-smooth, which is acceptable in applications. $\endgroup$ – Tony B Jun 11 '17 at 4:16
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Estimates for such sums can often be deduced from Deligne's version of the Riemann Hypothesis using computations of the relevant monodromy groups (typically by Katz, using sheaf-theoretic Fourier-transform arguments) and algebraic-group results (what Katz calls the Goursat-Kolchin-Ribet argument). For analytic number theory purposes, there's a survey called "A study in sums of products" (Fouvry, Kowalski, Michel) that you can find online, with examples and pointers to the literature.

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  • $\begingroup$ Thank you for the reference. I notice that Corollary 3.3 in that paper answers my question when $K$ is a Kloosterman sum. I'm not quite sure if the paper also considers the Weil's sum case. $\endgroup$ – Tony B Jun 10 '17 at 19:58
  • $\begingroup$ In this case, the relevant monodromy group might not be so easy to compute. I don't think Katz said much on this kind of sheaves. His student Ondrej Such did in his thesis, but I didn't find in it a clean statement of the kind you need for Goursat-Kolchin-Ribet. semanticscholar.org/paper/… $\endgroup$ – Will Sawin Jun 10 '17 at 20:26

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