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/
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