On the power moments of the Weil spectrum

Abstract

Let $\mathbb {F}$ be a finite field and let d be a positive integer with $\gcd \left( d, |\mathbb {F}^*|\right) =1$. Let $\psi$ be the canonical additive character of $\mathbb {F}$. The Weil spectrum associated with $\mathbb {F}$ and d is the set $\left\{ W_{\mathbb {F},d}(a): \ a\in \mathbb {F}^*\right\}$, where

$$\begin{aligned} W_{\mathbb {F},d}(a)=\sum _{x\in \mathbb {F}}\psi (x^d-ax). \end{aligned}$$

Let k be a non-negative integer. The k-th power moment of $W_{\mathbb {F},d}$ is defined by

$$\begin{aligned} P_{\mathbb {F},d}^{(k)}=\sum _{a\in \mathbb {F}^*}W_{\mathbb {F},d}(a)^k. \end{aligned}$$

In this paper we give exact identities of $P_{\mathbb {F},d}^{(k)}$ for $k=3,4,5,6$ at special cases.

Appendices

Appendix A: The Proof of Lemma 3.1

From the properties of residue systems we get

$$\begin{aligned} \Omega ^{(4,3)}&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_{2}\in \mathbb {F}} \sum _{x_{3}\in \mathbb {F}}\sum _{x_{4}\in \mathbb {F}}}_{x_1+x_{2}+x_3+x_4+1=0}}_{x_1^3+x_2^3+x_3^3+x_4^3+1=0}1 =\mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}}_{x_1+x_2+x_3=-1-x_4}}_{x_1^3+x_2^3+x_3^3=-1-x_4^3}1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}}_{x_1+x_2+x_3=-1-x_4}}_{(x_1+x_2+x_3)^3-(x_1^3+x_2^3+x_3^3)=(-1-x_4)^3-(-1-x_4^3)}1 \\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}} _{x_1+x_2+x_3=-1-x_4}} _{3(x_1+x_2)x_3^2+3(x_1+x_2)^2x_3+(x_1+x_2)^3-x_1^3-x_2^3=-3x_4(1+x_4)}1 \\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{3(x_1+x_2)x_3^2+3(x_1+x_2)^2x_3+(x_1+x_2)^3-x_1^3-x_2^3 =-3\left( 1+x_1+x_2+x_3\right) \left( x_1+x_2+x_3\right) }1\\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{3(x_1+x_2)x_3^2+3(x_1+x_2)^2x_3+(x_1+x_2)^3-x_1^3-x_2^3 =-3x_3^2-3(2(x_1+x_2)+1)x_3-3(x_1+x_2)^2-3(x_1+x_2)} 1\\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{3(x_1+x_2+1)x_3^2+3(x_1+x_2+1)^2x_3+(x_1+x_2+1)^3-x_1^3-x_2^3-1=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{x_1+x_2+1\ne 0}} _{36(x_1+x_2+1)^2x_3^2+36(x_1+x_2+1)^3x_3+12(x_1+x_2+1)^4-12(x_1+x_2+1)(x_1^3+x_2^3+1)=0} 1 +\mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{x_1+x_2+1=0}} _{x_1^3+x_2^3+1=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{x_1+x_2+1\ne 0}} _{\left( 6(x_1+x_2+1)x_3+3(x_1+x_2+1)^2\right) ^2+3(x_1+x_2+1)^4-12(x_1+x_2+1)(x_1^3+x_2^3+1)=0} 1 +|\mathbb {F}|\mathop {\sum _{x_2\in \mathbb {F}}} _{-(x_2+1)^3+x_2^3+1=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{x_1+x_2+1\ne 0}} _{x_3^2+3(x_1+x_2+1)^4-12(x_1+x_2+1)(x_1^3+x_2^3+1)=0} 1 +|\mathbb {F}|\mathop {\sum _{x_2\in \mathbb {F}}} _{x_2(x_2+1)=0} 1. \end{aligned}$$

Let $\eta$ denote the quadratic multiplicative character of $\mathbb {F}$. We have

$$\begin{aligned} \Omega ^{(4,3)}&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}} _{x_1+x_2+1\ne 0} \left( 1+\eta (12(x_1+x_2+1)(x_1^3+x_2^3+1)-3(x_1+x_2+1)^4)\right) +2|\mathbb {F}|\\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}} _{x_1+x_2+1\ne 0} \eta (12(x_1+x_2+1)(x_1^3+x_2^3+1)-3(x_1+x_2+1)^4) +|\mathbb {F}|^2-|\mathbb {F}| +2|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}^*}} \eta \left( 12y(x^3+(y-x-1)^3+1)-3y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}} \eta \left( 12y(-3x(x+1)+3(x+1)^2y-3(x+1)y^2)+9y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}} \eta \left( 4y(x+1)(-x+(x+1)y-y^2)+y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}} \eta \left( y(y-1)((2x-(y-1))^2-(y+1)^2)+y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( y(y-1)(z^2-(y+1)^2)+y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( y(y-1)(z^2-(y+1)^2)+y^4\right) +\mathop {\sum _{y\in \mathbb {F}}} \eta \left( -y(y-1)(y+1)^2+y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} (1+\eta (z))\eta \left( y(y-1)(z-(y+1)^2)+y^4\right) +\mathop {\sum _{y\in \mathbb {F}}} \eta \left( -y(y-1)(y+1)^2+y^4\right) \\&\quad + |\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( y(y-1)(z-(y+1)^2)+y^4\right) \\&\quad +\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( y(y-1)z^2+(y^4-y(y-1)(y+1)^2)z\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}} \eta \left( -y(y-1)(y+1)^2+y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( y(y-1)(z-(y+1)^2)+y^4\right) -\mathop {\sum _{y\in \mathbb {F}}} \eta \left( -y(y-1)(y+1)^2+y^4\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( y(y-1)+(y^4-y(y-1)(y+1)^2)z^{-1}\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}} \eta \left( -y(y-1)(y+1)^2+y^4\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{y(y-1)\ne 0} \eta \left( y(y-1)(z-(y+1)^2)+y^4\right) +\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{y(y-1)=0} \eta \left( y^4\right) \nonumber \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( y(y-1)+(y^4-y(y-1)(y+1)^2)z\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= |\mathbb {F}|+\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( y(y-1)+(y^4-y(y-1)(y+1)^2)z\right) +|\mathbb {F}|^2+|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{y^4-y(y-1)(y+1)^2\ne 0} \eta \left( y(y-1)+(y^4-y(y-1)(y+1)^2)z)\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{y^4-y(y-1)(y+1)^2=0} \eta \left( y(y-1)\right) +|\mathbb {F}|^2+2|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{y^4-y(y-1)(y+1)^2\ne 0} \eta \left( y(y-1)+(y^4-y(y-1)(y+1)^2)z\right) -\mathop {\sum _{y\in \mathbb {F}}} _{y^4-y(y-1)(y+1)^2\ne 0} \eta \left( y(y-1)\right) \\&\quad + (|\mathbb {F}|-1)\mathop {\sum _{y\in \mathbb {F}}} _{y^4-y(y-1)(y+1)^2=0} \eta \left( y(y-1)\right) +|\mathbb {F}|^2+2|\mathbb {F}|\\&= -\mathop {\sum _{y\in \mathbb {F}}} \eta \left( y(y-1)\right) +|\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}} _{y^4-y(y-1)(y+1)^2=0} \eta \left( y(y-1)\right) +|\mathbb {F}|^2+2|\mathbb {F}|\\&= -\mathop {\sum _{y\in \mathbb {F}^*}} \eta \left( 1-y\right) +|\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}^*}} _{y^2-y-1=0} \eta \left( y(y-1)\right) +|\mathbb {F}|^2+2|\mathbb {F}|\\&= 1+|\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}} _{(2y-1)^2=5} 1 +|\mathbb {F}|^2+2|\mathbb {F}| =|\mathbb {F}|^2+(3+\eta (5))|\mathbb {F}|+1. \end{aligned}$$

This proves Lemma 3.1.

Appendix B: The Proof of Lemma 3.2

It is not hard to show that

$$\begin{aligned} \Omega ^{(5,3)}&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}} \sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}\sum _{x_5\in \mathbb {F}}} _{x_1+x_2+x_3+x_4+x_5+1=0}}_{x_1^3+x_2^3+x_3^3+x_4^3+x_5^3+1=0}1 =\mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}\sum _{x_5\in \mathbb {F}}}_{x_1+x_2+x_3+x_4=-1-x_5}}_{x_1^3+x_2^3+x_3^3+x_4^3=-1-x_5^3}1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}\sum _{x_5\in \mathbb {F}}} _{x_1+x_2+x_3+x_4=-1-x_5}} _{(x_1+x_2+x_3+x_4)^3-(x_1^3+x_2^3+x_3^3+x_4^3)=(-1-x_5)^3-(-1-x_5^3)} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}\sum _{x_5\in \mathbb {F}}} _{x_1+x_2+x_3+x_4=-1-x_5}} _{3(x_1+x_2+x_3)x_4^2+3(x_1+x_2+x_3)^2x_4+(x_1+x_2+x_3)^3-(x_1^3+x_2^3+x_3^3)=-3x_5(1+x_5)} 1\\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}} _{3(x_1+x_2+x_3)x_4^2+3(x_1+x_2+x_3)^2x_4+(x_1+x_2+x_3)^3-(x_1^3+x_2^3+x_3^3)=-3(x_1+x_2+x_3+x_4)(1+x_1+x_2+x_3+x_4)} 1\\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}} _{3(x_1+x_2+x_3+1)x_4^2+3(x_1+x_2+x_3+1)^2x_4+(x_1+x_2+x_3+1)^3-(x_1^3+x_2^3+x_3^3+1)=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}} \sum _{x_4\in \mathbb {F}}} _{x_1+x_2+x_3+1\ne 0}} _{3(x_1+x_2+x_3+1)x_4^2+3(x_1+x_2+x_3+1)^2x_4+(x_1+x_2+x_3+1)^3-(x_1^3+x_2^3+x_3^3+1)=0} 1 +\mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}\sum _{x_4\in \mathbb {F}}} _{x_1+x_2+x_3+1=0}} _{x_1^3+x_2^3+x_3^3+1=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}\sum _{x_4\in \mathbb {F}}} _{x_1+x_2+x_3+1\ne 0}} _{36(x_1+x_2+x_3+1)^2x_4^2+36(x_1+x_2+x_3+1)^3x_4+12(x_1+x_2+x_3+1)^4-12(x_1+x_2+x_3+1)(x_1^3+x_2^3+x_3^3+1)=0} 1\\&\quad + |\mathbb {F}|\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}} _{x_1^3+x_2^3-(x_1+x_2+1)^3+1=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}\sum _{x_4\in \mathbb {F}}} _{x_1+x_2+x_3+1\ne 0}} _{(6(x_1+x_2+x_3+1)x_4+3(x_1+x_2+x_3+1)^2)^2+3(x_1+x_2+x_3+1)^4-12(x_1+x_2+x_3+1)(x_1^3+x_2^3+x_3^3+1)=0} 1\\&\quad + |\mathbb {F}|\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}} _{3(x_1+x_2)(x_1+1)(x_2+1)=0} 1\\&= \mathop {\mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}\sum _{x_4\in \mathbb {F}}} _{x_1+x_2+x_3+1\ne 0}} _{x_4^2+3(x_1+x_2+x_3+1)^4-12(x_1+x_2+x_3+1)(x_1^3+x_2^3+x_3^3+1)=0} 1 +|\mathbb {F}|(3|\mathbb {F}|-3). \end{aligned}$$

Let $\eta$ denote the quadratic multiplicative character of $\mathbb {F}$. From the properties of residue systems we have

$$\begin{aligned} \Omega ^{(5,3)}&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{x_1+x_2+x_3+1\ne 0} \left( 1+\eta (12(x_1+x_2+x_3+1)(x_1^3+x_2^3+x_3^3+1)-3(x_1+x_2+x_3+1)^4)\right) \\&\quad + 3|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{x_1\in \mathbb {F}}\sum _{x_2\in \mathbb {F}}\sum _{x_3\in \mathbb {F}}} _{x_1+x_2+x_3+1\ne 0} \eta \left( 12(x_1+x_2+x_3+1)(x_1^3+x_2^3+x_3^3+1)-3(x_1+x_2+x_3+1)^4\right) \\&\quad + |\mathbb {F}|^3-|\mathbb {F}|^2 +3|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( 12z(x^3+y^3+(z-x-y-1)^3+1)-3z^4\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( 12z(x^3+y^3+(z-x-y-1)^3+1)-3z^4\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4-4z(x+y+1)(x-z)(y+1-z)-4zy(y+1)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{x\in \mathbb {F}}\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4-z(y+1-z)((2x+y+1-z)^2-(y+1+z)^2)-4zy(y+1)\right) \\&\quad + |\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}}} \eta \left( z^4-z(y+1-z)(w^2-(y+1+z)^2)-4zy(y+1)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}^*}} \eta \left( z^4-z(y+1-z)(w^2-(y+1+z)^2)-4zy(y+1)\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}^*}} \left( 1+\eta (w)\right) \eta \left( z^4-z(y+1-z)(w-(y+1+z)^2)-4zy(y+1)\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}^*}} \eta \left( z^4-z(y+1-z)(w-(y+1+z)^2)-4zy(y+1)\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}^*}} \eta \left( -z(y+1-z)w^2+(z^4+z(y+1-z)(y+1+z)^2-4zy(y+1))w\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}}} \eta \left( z^4-z(y+1-z)(w-(y+1+z)^2)-4zy(y+1)\right) \\&\quad - \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\right) \\ \end{aligned}$$

$$\begin{aligned}&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}^*}} \eta \left( -z(y+1-z)+(z^4+z(y+1-z)(y+1+z)^2-4zy(y+1))w^{-1}\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}}} _{z(y+1-z)\ne 0} \eta \left( z^4-z(y+1-z)(w-(y+1+z)^2)-4zy(y+1)\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}}} _{z(y+1-z)=0} \eta \left( z^4-4zy(y+1)\right) \\&\quad + \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}^*}} \eta \left( -z(y+1-z)+(z^4+z(y+1-z)(y+1+z)^2-4zy(y+1))w\right) \\&\quad + |\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}\sum _{w\in \mathbb {F}}} _{z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\ne 0} \eta \left( -z(y+1-z)+(z^4+z(y+1-z)(y+1+z)^2\right. \\&\quad \left. -4zy(y+1))w\right) \\&\quad - \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)\ne 0} \eta \left( -z(y+1-z)\right) +|\mathbb {F}|\mathop {\sum _{z\in \mathbb {F}}} \eta \left( z^2(z-2)^2\right) \\&\quad + (|\mathbb {F}|-1)\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)=0} \eta \left( -z(y+1-z)\right) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= |\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{z^4+z(y+1-z)(y+1+z)^2-4zy(y+1)=0} \eta \left( -z(y+1-z)\right) -\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} \eta \left( -z(y+1-z)\right) \\&\quad + |\mathbb {F}|(|\mathbb {F}|-2) +|\mathbb {F}|^3+2|\mathbb {F}|^2-3|\mathbb {F}|\\&= |\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{(y+1)(z^2-(y+1)z-(y-1)^2)=0} \eta \left( z^2-(y+1)z\right) -\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( z^2-(y+1)z\right) +|\mathbb {F}|^3\\&\quad +3|\mathbb {F}|^2\\&\quad -5|\mathbb {F}|\\&= |\mathbb {F}|(\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{y+1=0} \eta \left( z^2\right) +\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{z^2-(y+1)z-(y-1)^2=0} \eta \left( (y-1)^2\right) -\mathop {\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{y+1=0}} _{z^2-(y+1)z-(y-1)^2=0} \eta \left( z^2\right) ) \\&\quad - \mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} \eta \left( 1-(y+1)z^{-1}\right) +|\mathbb {F}|^3+3|\mathbb {F}|^2-5|\mathbb {F}|\\&= |\mathbb {F}|(\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{z^2-(y+1)z-(y-1)^2=0} 1 -\mathop {\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}^*}} _{y-1=0}} _{z^2-(y+1)z-(y-1)^2=0} 1 -2) -\mathop {\sum _{z\in \mathbb {F}^*}\sum _{y\in \mathbb {F}}} \eta \left( -zy-z+1\right) \\ \end{aligned}$$

$$\begin{aligned}&\quad + |\mathbb {F}|^3+4|\mathbb {F}|^2-6|\mathbb {F}|\\&= |\mathbb {F}|(\mathop {\sum _{y\in \mathbb {F}}\sum _{z\in \mathbb {F}}} _{(2z-y-1)^2-(y+1)^2-4(y-1)^2=0} 1 -1) -\mathop {\sum _{z\in \mathbb {F}^*}\sum _{y\in \mathbb {F}}} \eta \left( -zy-z+1\right) +|\mathbb {F}|^3+4|\mathbb {F}|^2-9|\mathbb {F}|\\&= |\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}\sum _{w\in \mathbb {F}}} _{w^2-(5y^2-6y+5)=0} 1 +|\mathbb {F}|^3+4|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}} \left( 1+\eta (5y^2-6y+5)\right) +|\mathbb {F}|^3+4|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}} \eta \left( 5y^2-6y+5\right) +|\mathbb {F}|^3+5|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\mathop {\sum _{y\in \mathbb {F}}} \eta (5)\eta \left( (5y-3)^2+16\right) +|\mathbb {F}|^3+5|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\eta (5)\mathop {\sum _{y\in \mathbb {F}}} \eta \left( y^2+16\right) +|\mathbb {F}|^3+5|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\eta (5)(\mathop {\sum _{y\in \mathbb {F}^*}} \left( 1+\eta (y)\right) \eta \left( y+16\right) +1) +|\mathbb {F}|^3+5|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\eta (5)(\mathop {\sum _{y\in \mathbb {F}}} \eta \left( y+16\right) -1 +\mathop {\sum _{y\in \mathbb {F}^*}} \eta \left( 1+16y\right) +1) +|\mathbb {F}|^3+5|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|\eta (5)(\mathop {\sum _{y\in \mathbb {F}}} \eta \left( 1+16y\right) -1) +|\mathbb {F}|^3+5|\mathbb {F}|^2-10|\mathbb {F}|\\&= |\mathbb {F}|^3+5|\mathbb {F}|^2-(10+\eta (5))|\mathbb {F}|. \end{aligned}$$

This completes the proof of Lemma 3.2.