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Polynomial Diophantines: Independent Study for Integer Solutions

Date: 11/30/2011 at 13:36:55
From: Jake
Subject: Techniques on solving Diophantine equations 

I already know how to solve Diophantine equations of the form 

   ax^2 + by^2 + cx + dy + exy + f = 0

I would like to learn how to solve more polynomial Diophantine equations.
What kind of mathematics should I study -- integer points of elliptic
curves? or Thue equations? or ...?

Is it, for example, useful to study p-adic integers, or arithmetical
geometry, or algebraic geometry? I have studied Atiyah's and Macdonald's
"Commutative Algebra," but not more advanced methods. Book suggestions
would be nice!

My background is basics of advanced algebra, i.e., the basics of fields
and rings. I heard that Qing Liu's book of arithmetical geometry is fine,
but I found it a bit hard to understand how to use those methods to
determine integer points.



Date: 12/01/2011 at 16:37:06
From: Doctor Vogler
Subject: Re: Techniques on solving Diophantine equations 

Hi Jake,

Thanks for writing to Dr. Math.

There are many different types of Diophantine equations, and no method
that will solve all of them (that's Matiyasevich's theorem, which shows
that there is no solution to Hilbert's tenth problem). But many different
strategies will allow you to solve certain kinds of Diophantine equations.

Your quadratic example, for one, can be solved in a few different ways and
is intimately related to quadratic number fields. If you haven't studied
quadratic number fields, then I would recommend it. Indeed, I would
recommend studying number fields generally, if you are interested in
Diophantine equations, and if you haven't already done so.

Certain types of Diophantine equations are very related to arithmetic
geometry. You could consider this a subfield of algebraic geometry, which
might not seem to help you much in solving Diophantine equations -- and
yet ... strangely ... the shape of the set of complex solutions (i.e., the
genus of a curve) gives you a surprising amount of information about the
kinds of integer or rational solutions to the same equation. All of this
discussion also applies to elliptic curves, to about the same degree,
except that more is known about elliptic curves than higher-genus curves
and higher-dimensional surfaces, so you might get more results out of the
application of this knowledge.

The p-adic numbers can help somewhat, although a deep study of them might
be unnecessary. They mostly help in showing that there are no p-adic
solutions, and therefore can be no rational solutions. But usually it is
sufficient to use a simple mod-p argument (or at least mod a power of p).

Another subject that might interest you is Diophantine approximation,
which is essentially what "Transcendental Number Theory" is. This has to
do with Baker's Theorem and related theorems and methods. I think Thue
equations would fall into this category.

One book that comes to mind, especially for Diophantine approximation, is
"The Algorithmic Resolution of Diophantine Equations," by Nigel P. Smart.
I don't think I'm qualified to recommend a book on algebraic geometry, but
I think that Knapp's "Elliptic Curves" is a much more readable
introduction to elliptic curves than the more advanced (but more
comprehensive) "The Arithmetic of Elliptic Curves," by Silverman. So I'd
say to start with Knapp if you are new to them, but go to Silverman if you
have some knowledge and want to fill it out.

And of course, you can search our archives to see some strategies that we
have used to solve particular Diophantine equations that people have sent
us.

If you have any questions about this or need more help, please write back
and show me what you have been able to do, and I will try to offer further
suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 12/02/2011 at 11:16:57
From: Jake
Subject: Techniques on solving Diophantine equations 

Thanks.

While trying to solve the equation x^3 + 2y^3 = 3, I found this paper:
  http://www.ams.org/journals/mcom/2000-69-229/
S0025-5718-99-01124-2/S0025-5718-99-01124-2.pdf
  
It says I have to compute some subgroup of the unit group. But even the
unit group is a new term to me, so I need some background first.

I have Neukirch's book on algebraic number theory, but I'm not sure what
tricks in algebraic number theory are useful for Diophantine problems.

I found also the link
  http://math.stackexchange.com/questions/13507/third-degree-diophantine-
equation

I did not understand how to transform a curve to its Weierstrass form in
general, or how to compute Q|->Q(+)P, as mentioned in text. I should
probably start on quadratic number fields.

Thanks!



Date: 12/03/2011 at 00:47:40
From: Doctor Vogler
Subject: Re: Techniques on solving Diophantine equations 

Hi Jake,

For algebraic number theory and number fields in particular, I would
highly recommend the (old but very good) book "Number Fields," by Marcus.

Knapp's "Elliptic Curves" will talk about Weierstrass form, some elliptic
curve transformations, and certainly how to compute sums of points.

But perhaps a better resource for taking general degree-three curves (and
curves of the form y^2 = quartic in x) and transforming them into
Weierstrass form would be the online book "Elliptic Curve Handbook," by
Ian Connell. Section 1.4 shows how to do that conversion, and 1.2 does the
y^2 = quartic. But if you haven't already gone through an intro to
elliptic curves, such as Knapp, I should warn you that the book starts out
moving rather quickly.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Number Theory

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