On the estimate for the mixed 3-dimensional hyper-Kloosterman sum

Question

There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:

For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p} \left( \frac{z}{p}\right) e\left (\frac{\overline{z} x+ny+\overline{xy} }{p}\right)\ll_\varepsilon p^{\frac{3}{2}+\varepsilon}?$$ Here, $p$ is a prime and $\left( \frac{z}{p}\right)$ denotes the Jacobi symbol which is a Dirichlet character modulo $p$.

I searched P. Michel's book-Lectures on applied p-adic cohomology (https://arxiv.org/pdf/1712.03173v1.pdf), and E. Bombieri-S. Sperger's paper-On the estimation of certain exponential sums (Acta Arith. 69 (1995), 329-358). It seems that there is no a direct record regarding this type of sum with the character $\left( \frac{z}{p}\right)$ contained.

Perhaps, one can find the theory in, for example, L. Fu's paper-Weights of twisted exponential sums (Math. Z. 262 (2009), 449-472). However, it seems very difficult to discuss how the associated Laurent polynomial is non-degenerate with respect to its Newton polyhedron.

So, if any expert here know some knoledge on this question, please show some guides or corresponding references, many many thanks.

Thanks in advance!

Answer 1

This sum can be handled by elementary change of variables.

Let $a = \overline{z} x$, $b=y$, and $c =\overline{xy}$. Then $x = \overline{bc}$, $y=b$, and $z=\overline{abc}$, so $(x,y,z) \to (a,b,c)$ is an invertible change of coordinates that gives a permutation of the sum. So $$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p} \left( \frac{z}{p}\right) e\left (\frac{\overline{z} x+ny+\overline{xy} }{p}\right) = \sideset{_{}^{}}{^{\ast}_{}}\sum _{a,b ,c\bmod p} \left( \frac{abc }{p}\right) e\left (\frac{a+nb+c }{p}\right) .$$

This sum splits into sums over the individual variables $a,b,c$, which are all quadratic Gauss sums, so the size is exactly $p^{3/2}$ by a theorem of Gauss.