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Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.

For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$, how can we estimate the sum $$ \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod q} \left( \frac{x+\overline{xy}+\alpha y\overline{z}+\beta z+\gamma \overline{z}}{q}\right)?$$

I searched Pitt's paper(Theorem 3, https://sci-hub.wf/10.1215/S0012-7094-95-07711-4), Friedlander-Iwaniec's peper (Theorem2, https://sci-hub.wf/10.2307/1971175), and Michel's paper (Section 6, https://www.doc88.com/p-9052834716480.html?r=1), but find it seems they all don't match the type we are concerned above.

It can be sure some stuff involving $l$-adic cohomology can be put into use; see for example Adolphson-Sperber's paper (https://www.researchgate.net/profile/Steven-Sperber-2/publication/38390736_Exponential_sums_and_Newton_polyhedra/links/558c275708ae591c19d9efe8/Exponential-sums-and-Newton-polyhedra.pdf) or Denef-Loeser's paper (https://webusers.imj-prg.fr/~francois.loeser/inv91.pdf). However, I am not really familiar with the $l$-adic cohomology, and can not really figure out the Newton polyhedron $\Delta_\infty(f)$ of a Laurent polynomial $$f(x_1,x_2,x_3)=x_1+\overline{x_1x_2}+\alpha x_2\overline{x_3}+\beta x_3+\gamma \overline{x_3}\in \mathbf{F}_q[x_1,x_2,x_3, \overline{x_1x_2x_3}]$$ to verify whether or not $f$ is non-degenerate with respect to $\Delta_\infty(f)$.

So, if some expert has seen this type of sum in the question or leans something how to show the non-degenerateness, please give some comments or guide a reference.

Great thanks in advance!

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.

For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$, how can we estimate the sum $$ \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod q} \left( \frac{x+\overline{xy}+\alpha y\overline{z}+\beta z+\gamma \overline{z}}{q}\right)?$$

I searched Pitt's paper(Theorem 3, https://sci-hub.wf/10.1215/S0012-7094-95-07711-4), Friedlander-Iwaniec's peper (Theorem2, https://sci-hub.wf/10.2307/1971175), and Michel's paper (Section 6, https://www.doc88.com/p-9052834716480.html?r=1), but find it seems they all don't match the type we are concerned above.

It can be sure some stuff involving $l$-adic cohomology can be put into use; see for example Adolphson-Sperber's paper (https://www.researchgate.net/profile/Steven-Sperber-2/publication/38390736_Exponential_sums_and_Newton_polyhedra/links/558c275708ae591c19d9efe8/Exponential-sums-and-Newton-polyhedra.pdf) or Denef-Loeser's paper (https://webusers.imj-prg.fr/~francois.loeser/inv91.pdf). However, I am not really familiar with the $l$-adic cohomology, and can not really figure out the Newton polyhedron $\Delta_\infty(f)$ of a Laurent polynomial $$f(x_1,x_2,x_3)=x_1+\overline{x_1x_2}+\alpha x_2\overline{x_3}+\beta x_3+\gamma \overline{x_3}\in \mathbf{F}_q[x_1,x_2,x_3, \overline{x_1x_2x_3}]$$ to verify whether or not $f$ is non-degenerate with respect to $\Delta_\infty(f)$.

So, if some expert has seen this type of sum in the question or leans something how to show the non-degenerateness, please give some comments or guide a reference.

Great thanks in advance!

$\mathbf{\text{Edit:}}$ If $(\alpha,q)=1,$ one sees that the triple sum turns out to be $$q\sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x \bmod q} KL_2(\alpha \overline{x-\gamma}) KL_2(\beta x),$$ for which it seems there is still no record in the literature. This however can be compared with eq.(6.2) in https://www.doc88.com/p-9052834716480.html?r=1. But unfortunately, the available result is that for the type $\sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x \bmod q} KL_2(\alpha \overline{x-\gamma}) KL_2(\beta \overline{x})$.

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.

For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$, how can we estimate the sum $$ \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod c} \left( \frac{x+\overline{xy}+\alpha y\overline{z}+\beta z+\gamma \overline{z}}{q}\right)?$$

I searched Pitt's paper(Theorem 3, https://sci-hub.wf/10.1215/S0012-7094-95-07711-4), Friedlander-Iwaniec's peper (Theorem2, https://sci-hub.wf/10.2307/1971175), and Michel's paper (Section 6, https://www.doc88.com/p-9052834716480.html?r=1), but find it seems they all don't match the type we are concerned above.

It can be sure some stuff involving $l$-adic cohomology can be put into use; see for example Adolphson-Sperber's paper (https://www.researchgate.net/profile/Steven-Sperber-2/publication/38390736_Exponential_sums_and_Newton_polyhedra/links/558c275708ae591c19d9efe8/Exponential-sums-and-Newton-polyhedra.pdf) or Denef-Loeser's paper (https://webusers.imj-prg.fr/~francois.loeser/inv91.pdf). However, I am not really familiar with the $l$-adic cohomology, and can not really figure out the Newton polyhedron $\Delta_\infty(f)$ of a Laurent polynomial $$f(x_1,x_2,x_3)=x_1+\overline{x_1x_2}+\alpha x_2\overline{x_3}+\beta x_3+\gamma \overline{x_3}\in \mathbf{F}_q[x_1,x_2,x_3, \overline{x_1x_2x_3}]$$ to verify whether or not $f$ is non-degenerate with respect to $\Delta_\infty(f)$.

So, if some expert has seen this type of sum in the question or leans something how to show the non-degenerateness, please give some comments or guide a reference.

Great thanks in advance!

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.

For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$, how can we estimate the sum $$ \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod q} \left( \frac{x+\overline{xy}+\alpha y\overline{z}+\beta z+\gamma \overline{z}}{q}\right)?$$

I searched Pitt's paper(Theorem 3, https://sci-hub.wf/10.1215/S0012-7094-95-07711-4), Friedlander-Iwaniec's peper (Theorem2, https://sci-hub.wf/10.2307/1971175), and Michel's paper (Section 6, https://www.doc88.com/p-9052834716480.html?r=1), but find it seems they all don't match the type we are concerned above.

It can be sure some stuff involving $l$-adic cohomology can be put into use; see for example Adolphson-Sperber's paper (https://www.researchgate.net/profile/Steven-Sperber-2/publication/38390736_Exponential_sums_and_Newton_polyhedra/links/558c275708ae591c19d9efe8/Exponential-sums-and-Newton-polyhedra.pdf) or Denef-Loeser's paper (https://webusers.imj-prg.fr/~francois.loeser/inv91.pdf). However, I am not really familiar with the $l$-adic cohomology, and can not really figure out the Newton polyhedron $\Delta_\infty(f)$ of a Laurent polynomial $$f(x_1,x_2,x_3)=x_1+\overline{x_1x_2}+\alpha x_2\overline{x_3}+\beta x_3+\gamma \overline{x_3}\in \mathbf{F}_q[x_1,x_2,x_3, \overline{x_1x_2x_3}]$$ to verify whether or not $f$ is non-degenerate with respect to $\Delta_\infty(f)$.

So, if some expert has seen this type of sum in the question or leans something how to show the non-degenerateness, please give some comments or guide a reference.

Great thanks in advance!

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.

For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$, how can we estimate the sum $$ \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod c} \left( \frac{x+\overline{xy}+\alpha \overline{y}z+\beta z+\gamma \overline{z}}{q}\right)?$$

I searched Pitt's paper(Theorem 3, https://sci-hub.wf/10.1215/S0012-7094-95-07711-4), Friedlander-Iwaniec's peper (Theorem2, https://sci-hub.wf/10.2307/1971175), and Michel's paper (Section 6, https://www.doc88.com/p-9052834716480.html?r=1), but find it seems they all don't match the type we are concerned above.

It can be sure some stuff involving $l$-adic cohomology can be put into use; see for example Adolphson-Sperber's paper (https://www.researchgate.net/profile/Steven-Sperber-2/publication/38390736_Exponential_sums_and_Newton_polyhedra/links/558c275708ae591c19d9efe8/Exponential-sums-and-Newton-polyhedra.pdf) or Denef-Loeser's paper (https://webusers.imj-prg.fr/~francois.loeser/inv91.pdf). However, I am not really familiar with the $l$-adic cohomology, and can not really figure out the Newton polyhedron $\Delta_\infty(f)$ of a Laurent polynomial $$f(x_1,x_2,x_3)=x_1+\overline{x_1x_2}+\alpha \overline{x_2}x_3+\beta x_3+\gamma \overline{x_3}\in \mathbf{F}_q[x_1,x_2,x_3, \overline{x_1x_2x_3}]$$ to verify whether or not $f$ is non-degenerate with respect to $\Delta_\infty(f)$.

So, if some expert has seen this type of sum in the question or leans something how to show the non-degenerateness, please give some comments or guide a reference.

Great thanks in advance!

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.

For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$, how can we estimate the sum $$ \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod c} \left( \frac{x+\overline{xy}+\alpha y\overline{z}+\beta z+\gamma \overline{z}}{q}\right)?$$

I searched Pitt's paper(Theorem 3, https://sci-hub.wf/10.1215/S0012-7094-95-07711-4), Friedlander-Iwaniec's peper (Theorem2, https://sci-hub.wf/10.2307/1971175), and Michel's paper (Section 6, https://www.doc88.com/p-9052834716480.html?r=1), but find it seems they all don't match the type we are concerned above.

It can be sure some stuff involving $l$-adic cohomology can be put into use; see for example Adolphson-Sperber's paper (https://www.researchgate.net/profile/Steven-Sperber-2/publication/38390736_Exponential_sums_and_Newton_polyhedra/links/558c275708ae591c19d9efe8/Exponential-sums-and-Newton-polyhedra.pdf) or Denef-Loeser's paper (https://webusers.imj-prg.fr/~francois.loeser/inv91.pdf). However, I am not really familiar with the $l$-adic cohomology, and can not really figure out the Newton polyhedron $\Delta_\infty(f)$ of a Laurent polynomial $$f(x_1,x_2,x_3)=x_1+\overline{x_1x_2}+\alpha x_2\overline{x_3}+\beta x_3+\gamma \overline{x_3}\in \mathbf{F}_q[x_1,x_2,x_3, \overline{x_1x_2x_3}]$$ to verify whether or not $f$ is non-degenerate with respect to $\Delta_\infty(f)$.

So, if some expert has seen this type of sum in the question or leans something how to show the non-degenerateness, please give some comments or guide a reference.

Great thanks in advance!