FWF - DioPro - Diophantine Problems: Analytic, geometric and computational aspects

Within this project various diophantine problems are investigated by classical tools from diophantine approximation as well as by modern methods from diophantine geometry. Furthermore, transcendence questions, effective methods and number-theoretic algorithms as well as computational aspects are included. Topics of special interest are the Andr-Oort conjecture, Baker's effective method of linear forms in logarithms of algebraic numbers and Schanuel's conjecture. In particular, in cooperation with G. Wüstholz we will contribute to questions which arise from a motivic view onto transcendence. This viewpoint has been suggested by work of Kontsevich and Zagier and Kontsevich's conjecture is deep and far-reaching. A further focus lies on the so-called Leibniz conjecture and its connection with transcendence theory. Another part of the project is related to polynomial decomposition theory. Since the work of J.F. Ritt in the 1920's this development was heavily influenced by several authors, in particular by M. Fried and in 2000 Y. Bilu and R.F. Tichy succeeded in fully combining polynomial decomposition theory with the classical theorem of Siegel on finiteness of integral points on curves of genus greater than 0, to give a complete ineffective criterion on the finiteness of the number of integer solutions x,y of Diophantine equations of the form f(x)=g(y). Within the frame of this project we want to further explore this kind of questions. In particular we will study certain invariants introduced by Blardon and Ng as well as by recent work of Müller and Zieve. This heavily involves a detailed study of monodromy groups of polynomials. More explicit investigations are devoted to applications of Baker's method to specific families of diophantine equations and of methods from diophantine approximation. By the famous result of Matijasevitch on Hilbert's 10th problem there is no algorithm for the solution of a general polynomial equation F(x1,...,xk)=0 in integers x1,...,xk. Thus the development of algorithms for the solution of more special diophantine equations is of great importance and remarkable progress in this field has been made in the last decades. Within this project we include applications to polynomial-exponential equations and to diophantine m-tuples. Furthermore, we address a specific problem involving continued fractions and various aspects of lattice point counting and estimates for heights. These techniques are useful in the investigation of the arithmetic complexity of algebraically defined objects. Whereas in diophantine geometry heights have become an important tool, the use of heights in group theory and additive combinatorics is a rather new development. Next fall semester a special semester at the ESI (Vienna) organized by R. Tichy, J. Vaaler, M. Widmer and U. Zannier is devoted to the interplay of these fields. Hopefully, this is also the starting point of this project.