From: Jan-Christoph Puchta <jcp@arcade.mathematik.uni-freiburg.de>
Subject: Re: "Squarefree Polynomials"
Date: Fri, 14 Jul 2000 10:51:04 +0200
Newsgroups: sci.math
Summary: [missing]

Dr. Michael Ulm wrote:

> On Thu, 13 Jul 2000 17:32:36 GMT, Bob Silverman <bobs@my-deja.com> wrote:
> >In article <Pine.LNX.4.10.10007130927200.9507-
> >100000@dl2sql9.princeton.edu>,
> >  Jan Kristian Haugland <haugland@dl2sql9.princeton.edu> wrote:
> >>
> >> > What is the fraction of polynomials, with integer coefficients, that
> >> > are not divisible by any square of a polynomial
> >>
> >> Depends on how you list the polynomials.
> >
> >The general conanonical way of doing this ordering is by discriminant.
> >With this ordering, the set of irreducible polynomials has density 1.
> >The answer follows.
> >
>
> However, if we restrict ourselves to polynomials of order <= d, then
> we can give a different answer. If we examine the fraction of all
> square-free polynomials with absolute value of every coefficient <= n,
> then it is not too hard to see, that this fraction converges to the
> fraction of d-tupels of Integers that are all divisible by the same
> square. In other words, the set of polynomials of order <= d that are
> not divisible by a square of a polynomial with degree > 0 has density
> 1.
>
> It rests to find the fraction of d-tupels of Integers that are
> divisible by the same square > 1. I have not slept enough this night
> (sick child) to find the answer to this right now. Shouldn't be too
> hard 'though.

The probability that d integers have the coomon divisor n is n^{-d}.
Applying Inclusion-Exclusion and counting rounding errors we get for N(x,
d), i.e. the number of integral d-tupels with squarefree GCD the asymptotic

N(x, d) = x sum_{n=1}^infinity mu(n) n^{-2d} = x/zeta(2d) + O(x^c(d))

with c(d)<1 and c(d) -> 0 for d -> infinity, hence the proportion of
nonsquarefree polynomials tends to 1-1/zeta(2d).

However, these common divisors might be called trivial. I would call a
polynomial like 4x^2+4 squarefree, and then almost all poynomials with
integer coeficients are squarefree.

In general, over a field F one should have proportion of squarefree
polynomials = 1 <==> F is finite, however, it seams to be difficult to
define the word "almost all" for an abstract field. The same should be true
for integral domains. However, if the base ring has zero divisors or is
noncommutative, polynomials have much more factors than over fields, and
their probability to be squarefree should decrease. However, a general
theory may become quiet dificult.


Jan-Christoph Puchta