Exercise 8.23

Proof. Let $A_r=\{(x_1,x_2,\cdots ,x_n)\in {𝔽}_p^n|f(x_1,x_2,\cdots ,x_n)=r\}$. Then ${𝔽}_p^n={\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{r\in {𝔽}_p}{A}_r$, so, for all $a\in {𝔽}_p$,

Let $m(r)=|A_r|=N(f(x_1,x_2,\cdots ,x_n)=r)$. Then

As $\underset{a\in {𝔽}_p}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{\zeta }^{\mathit{ar}}=0$ if $r\ne 0$, and $\underset{a\in {𝔽}_p}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{\zeta }^{\mathit{ar}}=p$ if $r=0$, we obtain

Moreover

so

In conclusion,

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