Linear forms in logarithms and Diophantine analysis (60 hours)

Course description

In this course we will describe in details application of linear forms in logarithms of algebraic numbers to Diophantine problems.

The introduction will cover some results from Diophantine approximations and algebraic number theory that we will use later in solving Diophantine equations.

Main part of the course will study linear forms as one of the modern method for solving Diophantine equations. Linear form in logarithms of algebraic numbers is an expression of the form where are algebraic numbers and are integers. From Baker's theory (theorems of Baker-Wustholz and Matveev) we get a lower bound for this form. For applications of linear forms to solving Diophantine equation the strategy is following. We first associate 'large' solutions of Diophantine equation to a 'very small value' of specific linear form in logarithms, which gives us an upper bound for the values of this linear form corresponding to a solution of the equation. After that, comparing this upper bound with the lower bound from Baker's theory, we get an upper bound for the values of the unknowns of our Diophantine equation. Usually this upper bound will be very large, so we will describe some methods from Diophantine approximations (continued fractions and LLL-algorithm) that will reduce this bound. We will describe those methods in details on some problems. The problems will include some equations with recurrence sequences, systems of simultaneous pellian equations, Catalan equation and Thue equations.

It will be assumed that the students are familiar with the basic notions and results from number theory.

References

  1. H. Cohen: Number Theory. Volume I: Tools and Diophantine Equations, Springer Verlag, Berlin, 2007.
  2. H. Cohen: Number Theory. Volume II: Analytic and Modern Tools, Springer Verlag, Berlin, 2007.
  3. A. Baker, G. Wustholz: Logarithmic Forms and Diophantine Geometry, Cambridge University Press, Cambridge, 2008.
  4. I. Gaal: Diophantine Equations and Power Integral Bases, Birkhauser, Boston, 2002.
  5. W. M. Schmidt: Diophantine Approximation and Diophantine Equations, Springer-Verlag, Berlin, 1996.
  6. T. H. Shorey, R. Tijdeman: Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986.
  7. N. P. Smart: The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998.
  8. V. G. Sprindžuk: Classical Diophantine Equations, Springer, Berlin, 1993.
  9. J. Steuding: Diophantine Analysis, Chapman & Hall/CRC, Boca Raton, 2005.
  10. B. M. M. de Weger: Algorithms for Diophantine Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.
  11. G. Wustholz (Ed.): A Panorama of Number Theory or The View from Baker'sGarden, Cambridge University Press, Cambridge, 2002.
  12. U. Zannier: Some applications of Diophantine Approximation to Diophantine Equations, Forum Editrice, Udine, 2003.