From: mdekker@fwi.uva.nl (Martijn Dekker)
Newsgroups: sci.math
Subject: zeta functions of schemes
Date: 17 Jan 1995 16:00:12 +0100

Hi,

just for fun I am writing a program that calculates (approximates)
the zeta function of a curve in projective 3-space, defined over
a finite field Fp.
To do this one needs to count points on the curve C. First count
all points with coordinates in Fp, call this number N_1, then
count the points with coordinates in Fp^2, call this number
N_2 etc. Now the zeta function of C is defined as

 Z(t) = exp(N_1 * t + (N_2)/2 * t^2 + (N_3)/3 * t^3 + ... )

Of course, to count points on Fp^i one needs a representation
of this field. I use Fp^i = Fp[X]/f, where f is an irreducible
polynomial in Fp[X] of degree i.

The problem is: how to find such an f. Of course I could use symbolic
packages (such as Maple, Mathematica or Reduce) to do this, but I like 
to do some real programming and I have written it in C.
The program stores the polynomials it finds in a database, so he only
starts searching for a polynomial of given degree over a given Fp,
when such a polynomial is not already in the database. But still, I 
would like to make this 'search for f' routine as fast as possible.
The obvious tricks to find factors of polynomials I already implemented
(looking for zeros etc). 
My question: anyone knows of clever tricks to see if a polynomial cannot
be irreducible (or must be irreducible) ?
The program to find an f is already up and running but I would like
to optimize it.
Now I am writing the code of actually counting the points...

If anyone is interested in this program (when it is finished) just
send me mail, and I will send you a copy. It is plain ANSI C and
runs on ANY computer (assuming you have a ANSI C Compiler)

-- 
Martijn Dekker             +---      Ishido by PFF software        ---+
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