From - Wed Jul 16 11:14:14 1997
From: elkies@ramanujan.math.harvard.edu (Noam Elkies)
Newsgroups: sci.math.research
Subject: Re: Generating function question
Date: 15 Jul 1997 21:13:27 GMT
Organization: Harvard Math Department
Summary: Sorry, it's transcendental
In article <5qdt5j$2tp@tierra.santafe.edu>,
Cris Moore wrote:
>Consider the generating function of the mod-2 Pascal's triangle,
>with variables x and y indicating the position of each term:
> 1
>+y +xy
>+y^2 +x^2y^2
>+y^3 +xy^3 +x^2y^3 +x^3y^3
>+y^4 +x^4y^4
>...
>
>clearly this is algebraic in Z_2. Is it algebraic in Z?
>This would be helpful to me in solving a statistical mechanics problem.
It is not even algebraic in C(x,y). This function has the infinite
product expansion
(1+x+y)(1+x^2+y^2)(1+x^4+y^4)(1+x^8+y^8)(1+x^16+y^16)...
in a neighborhood of (0,0), and thus by analytic continuation
on its polydisc of convergence |x|<1, |y|<1. In particular,
for n=0,1,2,... the function vanishes on the portion of the curve
x^(2^n)+y^(2^n)+1=0 contained in that polydisc. But it is not
possible for an algebraic function to vanish on infinitely many curves.
--Noam D. Elkies (elkies@math.harvard:edu)
Dept. of Mathematics, Harvard University
From - Wed Jul 16 21:30:14 1997
From: Robin Chapman
Newsgroups: sci.math.research
Subject: Re: Generating function question
Date: Tue, 15 Jul 1997 14:56:02 GMT
Organization: University of Exeter
Cris Moore wrote:
>
> Consider the generating function of the mod-2 Pascal's triangle,
> with variables x and y indicating the position of each term:
>
> 1
> +y +xy
> +y^2 +x^2y^2
> +y^3 +xy^3 +x^2y^3 +x^3y^3
> +y^4 +x^4y^4
> ...
>
> clearly this is algebraic in Z_2. Is it algebraic in Z?
> This would be helpful to me in solving a statistical mechanics problem.
>
This has a nice infinite product, i.e., it's the product of
1 + y^(2^j) + (xy)^(2^j) over all integers j >= 0. It is not
algebraic though. Calling this sum f(x, y), then g(y) = f(1, y)
is the sum of 2^{d(m)} x^m where d(m) is the number of ones
in the binary expansion of m. If f is algebraic, then g is too.
But 2^{d(m)} <= m + 1 for all m , and so g is analytic in the
open unit disc. But g(y) = (1 + 2y)(1 + 2y^2)(1 + 2y^4)...
has infinitely many zeros in the open unit disc --- impossible
for a rational function.
--
Robin Chapman "256 256 256.
Department of Mathematics O hel, ol rite; 256; whot's
University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no.
rjc@maths.exeter.ac.uk 2 dificult 2 work out."
http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn