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First properties of congruent Zeta function

Let us first recall an elementary formula

LEMMA 3.2  

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

DEFINITION 3.3   We denote by $ \mathbb{A}_n$ the ``void set of equation'' in $ n$ -variables. That means, for any field (or ring) $ k$ , we put

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

PROPOSITION 3.4  

% latex2html id marker 784
$\displaystyle Z(\mathbb{A}_n/\mathbb{F}_q,T)= \frac{1}{1-q^n T}
$

PROPOSITION 3.5   Let $ V,W,W_1,W_2$ be sets of equations.
  1. If % latex2html id marker 793
$ \char93 V(\mathbb{F}_{q^s})= \char93 W(\mathbb{F}_{q^s})$ for any $ s$ ,then % latex2html id marker 797
$ Z(V/\mathbb{F}_q,T)
=Z(W/\mathbb{F}_q,T)$ .
  2. If % latex2html id marker 799
$ \char93 V(\mathbb{F}_{q^s})= \char93 W_1(\mathbb{F}_{q^s})+\char93 W_2(\mathbb{F}_{q^s})$ for any $ s$ ,then:

    % latex2html id marker 803
$\displaystyle Z(V/\mathbb{F}_q,T) =Z(W_1/\mathbb{F}_q,T) Z(W_2/\mathbb{F}_q,T).
$

PROPOSITION 3.6   Let % latex2html id marker 810
$ 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. \begin{displaymath}
% latex2html id marker 816V(\mathbb{F}_{q^s})=
\begin{cases}
d & \text {if } d \vert s \\
0 & \text {otherwise}
\end{cases}\end{displaymath}

  2. % latex2html id marker 818
$\displaystyle Z(V/\mathbb{F}_q,T) = \frac{1}{1-T^d}
$

EXERCISE 3.3   Describe what happens when we omit the assumption of $ f$ being irreducible in Proposition 3.6.


next up previous
Next: About this document ... Up: Zeta functions. No.3 Previous: Definition of congruent Zeta
2013-04-24