Diploma Thesis  

Algorithmic Solution of Diophantine Equations  (Stoll Thomas) (ps)
 
The Diophantine equation X^M-Y^N=1, known as Catalan's equation, can well be treated with Baker's theory of linear forms in logarithms. In detail, we give both Baker's proof of his general transcendence result and Tijdeman's proof on Catalan's equation, which gives an explicit upper bound for max(x,y,m,n). This method for showing effective finiteness also works for elliptic and hyperelliptic equations. In the thesis much emphasize is laid on the decidability of non-effective finiteness of the more general Diophantine equation F(X)=G(Y), F and G being arbitrary polynomials. Recent progress has been made by Bilu and Tichy in showing that, by making explicit Siegel's theorem, these
polynomials have to satisfy polynomial identities with the so-called standard pairs. In the sequel, an algorithmic implementation of such criterion is given as a Maple worksheet. We study the case, where F and G belong to a polynomial family. For this reason we include a general approach to define orthogonal polynomials and state the fundamental properties they all have in common. In the case where F and G are Meixner polynomials of fixed degree, we prove that there are only finitely many integral solutions (X,Y) to F(X)=G(Y). Geometrically speaking, two octahedrons of dimensions N and M, respectively, can have equally many integral points just in a finite number of cases.
 
thesis advisor: Tichy Robert, Dr.phil., O.Univ.-Prof.
organization: Working Group Mathematics A of the Institute of Mathematics
year of publication: 2001