First properties of congruent Zeta function

Let us first recall an elementary formula

LEMMA 4.1  

$$\sum_{k=1}^\infty \frac{1}{k} T^k = -\log(1-T)$$

DEFINITION 4.2   We denote by $\mathbb{A}_n$ the ``void set of equation'' in $n$ -variables. That means, for any field (or ring) $k$ , we put

$$\mathbb{A}_n(k)=\{ x\in k^n\}.$$

PROPOSITION 4.3  

$$Z(\mathbb{A}_n/\mathbb{F}_q,T)= \frac{1}{1-q^n T}$$

PROPOSITION 4.4   Let $V,W,W_1,W_2$ be sets of equations.

1. If $\char93 V(\mathbb{F}_{q^s})= \char93 W(\mathbb{F}_{q^s})$ for any $s$ ,then $Z(V/\mathbb{F}_q,T) =Z(W/\mathbb{F}_q,T)$ .
2. If $\char93 V(\mathbb{F}_{q^s})= \char93 W_1(\mathbb{F}_{q^s})+\char93 W_2(\mathbb{F}_{q^s})$ for any $s$ ,then:
   $$Z(V/\mathbb{F}_q,T) =Z(W_1/\mathbb{F}_q,T) Z(W_2/\mathbb{F}_q,T).$$

PROPOSITION 4.5   Let $f\in \mathbb{F}_q[X]$ be an irreducible polynomial in one variable of degree $d$ . Let us consider $V=\{f\}$ , an equation in one variable. Then:

1. $$% latex2html id marker 647V(\mathbb{F}_{q^s})= \begin{cases} d & \text {if } d \vert s \\ 0 & \text {otherwise} \end{cases}$$

2. $$Z(V/\mathbb{F}_q,T) = \frac{1}{1-T^d}$$

EXERCISE 4.1   Describe what happens when we omit the assumption of $f$ being irreducible in Proposition 4.5.