From: Roberto Maria Avanzi
Subject: Re: polynomial pell's equations
Date: Fri, 26 Feb 1999 11:59:50 +0100
Newsgroups: sci.math.research
Keywords: Solution to X^2 - D Y^2 = 1 in polynomials D,X,Y
On Tue, 23 Feb 1999, James G. McLaughlin wrote:
> At the end of a paper in the Proceedings of the AMS (vol 56 1976 p.89 -
> 92) Melvyn Nathanson adds the comment that David Zeitlin had communicated
> to him personally the observation that the solutions to the polynomial
> Pell's equations can all be expressed neatly in terms of the Chebyshev
> polynomials T_n(x) and U_n(x).
> Does anyone know of a reference for the connection between
> solutions to the polynomial Pell's equations and these Chebyshev
> polynomials T_n(x) and U_n(x)?
>
> (A solution to the polynomial Pell's equation is a triple of polynomials
> c(t), f(t) and h(t) satisfying c(t)^2 - f(t)h(t)^2 = 1.)
>
Let's consider characteristic 0.
Embed everything in C.
Uhm... if f(t) is a squarefree degree 2 polynomial then
you get essentially the Chebyshev polynomials composed
with linear functions. You can assume you want to
solve it with f(t)=t^2-4, then if you look at Schinzel's
book "Selected topics in polynomials" you will find
the connection. These is also an enlightening paper
by Dorey and Whaples around about Ritt's 2nd theorem,
and the contributions of Michael Fried to the subject
are far reaching. In this case, the solution to the
problem is very well known.
If f(t) is a square, assume f(t)=t^2 and you get no
solutions apart from constants.
f(t) must have anyway have even degree.
A recent work by Zannier and myself had to consider
the case when f(t) has degree 4. It arises naturally
in a diophantine context. then we can assume f(t)
is square free, otherwise we simply "push" the
squared factor inside h(t). You get something more
than simply Chebishev polynomials, and you parametrize
the solutions via modular curves.
Cheers
Roberto
PS: just email me privately: what is your research about ?
who's your supervisor ?
> Jimmy Mc Laughlin,
> UIUC,
> Champaign, Illinois.
[deletia -- djr]
_/_/ Roberto Maria Avanzi
/_/ Institut für Experimentelle Mathematik / Universität Essen
_/ Ellernstraße 29 / 45326 Essen / Germany
/ Phone: ++49-201-183-7641 Fax: ++49-201-183-7668
==============================================================================
From: Allan Adler
Subject: Re: polynomial pell's equations
Date: 27 Feb 1999 23:54:03 -0500
Newsgroups: sci.math.research
Abel wrote a paper relevant to this question. It appeared in
Crelle, vol.1, in 1826. He wanted to know when, for polynomials M,N,R in x,
the integral of (M dx)/(N sqrt(R)) can be expressed in the form
log ( (p+q sqrt(R))/(p-q sqrt(R))) where p,q are polyomials in x.
This leads him to study the equation p^2-q^2 R = N, which he then
reduces to the study of the case N=1. He then pursues this to obtain
a continued fraction which he studies in some detail. In the last
section, he works out the special case of elliptic integrals. At the
end of the paper, he announces that if rho is a polynomial in x and if
the integral of (rho dx)/sqrt(R) is expressible in any way at all using
logarithms, then it is expressible in the form
A log ( (p+q sqrt(R))/(p-q sqrt(R)))
where A is a constant and p,q are polynomials, and promises to prove
it on another occasion.
Allan Adler
ara@altdorf.ai.mit.edu