Re: [Phys-l] diophantine equations

[John Denker <jsd@av8n.com>]

On 01/15/2011 10:09 AM, Stefan Jeglinski wrote:
> Why are diophantine equations defined as having integer-only
> solutions? Is it merely so that those solutions that are integer-only
> have a name? 

Yes.

> It seems easy to find a "diophantine equation" whose
> solutions are real or complex, as well as integer. 

No; see below.

> Is it then no
> longer a diophantine equation?

Yes, that's the point.

> I guess I'm trying to get at whether this is just a semantic
> distinction or whether there is a deeper reason for the naming
> convention. 

If you are pointing out that the name is not very descriptive,
I'd say that is true but not worth worrying about.

By way of analogy:  A titmouse is not a mouse.  Buckwheat is not 
a kind of wheat.  As Voltaire pointed out, the Holy Roman Empire 
was neither holy, nor Roman, nor an empire.

The point is that it is extremely common to find names that are
not descriptive.  Here is a collection of a few dozen particularly
weird examples:
   http://www.av8n.com/physics/weird-terminology.htm

Since a Diophantine equation is in fact an equation, I would say
this is actually a rather mild case of the disease.

  Suggestion:  If "Diophantine equation" really bugs you, one option 
  is to speak of "Diophantine analysis" instead.  That terms is fully 
  accepted and is almost as widely used.

More generally, though, you should not expect names to be descriptive.
Even if you have a chance to give something a descriptive, name, I
recommend that you do not.  For example, when naming the variables
and the subroutines in a computer program, oftentimes it would take
several sentences or several paragraphs to fully describe what the 
thing does ... and that's too long for a convenient name.  It is 
better to give it a short name and then write into the documentation
a _legend_ that says what the name means.