To: David Rusin <rusin@mp.cs.niu.edu>
Subject: Re: seeking rational functions as group operations
Date: Wed, 10 Nov 93 13:40:14 -0800
From: "David G. Cantor" <dgc@ccrwest.org>

In your posting you say:

      I am looking for examples of rational functions which define an
      associative binary operation on (most of) a vector space (over the
      rationals, say).  For example on a one-dimensional vector space
      (Q itself) we have the two polynomials F(x,y)=x+y and F(x,y)=xy
      which define the group operations on (Q,+) and (Q^*, \times).  But
      it is a typical exercise to show that F(x,y) = x+y+xy is also
      associative [using x-> x+1 we see that this is the same group
      as Q^*].  Similarly, F(x,y)=(xy)/(x+y) works almost everywhere
      [using x -> 1/x we recognize the group (Q,+)] although there is a
      difficulty when x=-y which I am happy to ignore. . .
------------------------------------------------------------------------
Somethings that come to mind are addition on elliptic curves.  There are
numerous books in the literature on how to add on such a curve.  There
are also group laws on the Jacobians o
hyperelliptic curves.  These are curves  given by equations of the
form

$Y^2=F(X)$ where $F(X)$ is a monic poynomial of degree $2g+1$ ($g$ is
called the genusa.

I published a paper giving the explicit addition law in Math. of Comp.
a few years ago.

dgc
 
David G. Cantor
Department of Mathematics
University of California
Los Angeles, CA 90024
dgc@math.ucla.edu