From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: Analytic function selecting ZxZ
Date: 14 Feb 1999 13:46:34 -0500
Newsgroups: sci.math
Keywords: Application of Weierstrass Preparation Theorem

vmhjr@frii.com (Virgil) writes:

>In article <7a67vs$2e8@elk.cs.sfu.ca>, wehner@cs.sfu.ca (Stephan Wehner) wrote:
>
>>Hi there,
>>
>>is there an analytic function f:CxC -> C, so that f(x,y) = 0 if and only if
>>x and y are integer?
>>
>>Stephan
>>
>>(please send me email at wehner@cs.sfu.ca when responding)
>
>
>Will f(x,y) = sin(x/pi)^2 + sin(y/pi)^2 do?

No (assuming, as I think we must, that SW means "complex analytic" by
"analytic"), and not just because of that unfortunate accident which
tore two limbs off each of the times signs and left them looking like
division signs, nor because of the more serious problem that the sum 
of the squares of two or more non-zero complex numbers can easily be 0. 

In fact, a non-constant complex analytic function from CxC to C
is going to vanish on a complex-analytic curve, which (as a space)
has (real) dimension 2 at each of its points; clearly ZxZ isn't
such a curve.  A sledgehammer proof would invoke the Weierstrass
Preparation theorem.

Lee Rudolph