From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Minimal Forms of Diophantine Equations
Date: 21 Aug 1995 06:57:46 GMT

In article <40te57$2co@kaleka.seanet.com>,
Kevin Brown <ksbrown@ksbrown.seanet.com> asked a number of questions
regarding canonical forms for Diophantine equations. It is difficult to
summarize the possible answers to these questions, but I would like to
caution that one needs to think carefully about what "solutions" and
"equivalence classes" are supposed to mean. For example, the equations
x^2+1=0 and x^2+2=0 are not equivalent in any usual sense but they
have the same solution set (over the integers!). Given any collection of
pairs (say) of integers (say), one can look at the ideal of all polynomials
in in  Z[x,y] (say) which vanish at all these points; if it's principal, then
a generator would be a reasonable candidate for the canonical form of
a polynomial which defined the collection in the first place, but there's no
guarantee of principality, nor does this perspective consider distinct
collections of points to be "the same", which your questions suggest you
would want to do. In short, it makes the most sense to begin with a
family of algebraic sets which you expect to have some similar structure
(e.g. conics) and then define a canonical form for them.

>More generally, is it possible to define a unique "minimal reduced
>form" for other kinds of Diophantine equations (not restricted to
>elliptic curves) such that if each solution of equation A corresponds
>to a solution of equation B, then the reduced forms of A and B are
>identical?

Typically one asks that the solution sets be more than just of the same
cardinality; there should be for example a linear map of the ambient space
carrying one solution set to the other. But there are many choices one
could make, leading to different discussions.

dave