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 André-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.

Diophantine problems are mathematical problems that have to be solved in integers. In particular, these are equations or inequalities, some of which relate to classical problems of interest since antiquity. For example, the study of Diophantine m-tuples goes back to Diophantus of Alexandria. Significant progress has been made on this topic in recent years, and this project has made important contributions. The project has also significantly stimulated international cooperations, which has led to a broad methodological scope and the involvement of guest scientists and postdocs. Particularly noteworthy is the awarding of an honorary professorship of Graz University of Technology to Gisbert Wüstholz and the honorary doctorate from the University of Debrecen to Robert Tichy. Several young scientists were also involved in this project, in particular I would like to mention Mrs. Kreso and Mrs. Chim. Both of them did their doctorate at Graz University of Technology on Diophantine Problems. In her dissertation, Mrs. Kreso dealt with decomposability properties of polynomials and applications to Diophantine equations. She received a "best paper award" for her work and a Schrödinger FWF scholarship, which she spent in Canada. She is already an internationally respected young scientist currently spending her Schrödinger return phase in Graz. Mrs. Chim was supervised jointly by G. Wüstholz and R. Tichy. She is originally from Hong Kong and has continued the work of Kunrui Yu on p-adic linear forms in logarithms. She, too, has made an international appearance, for example by a stay at the Max Planck Institute in Bonn. The results of this project have been published in world-class international journals and open access platforms (e.g., arxiv). In addition, the members of the project have given a variety of lectures at international conferences on the results achieved. The project leader was a keynote speaker at symposia and workshops on several occasions. In the final phase of the project, another PhD student, Mahadi Ddamulira, was successfully recruited to work on Diophantine problems. In summary, this project has provided an excellent working environment for young, internationally recruited researchers and has opened the door to a promising academic career. The cooperation of these young researchers with established personalities and international guests has led to significant advances in Diophantine number theory, such as Diophantine m-tuples, polynomial decomposability and polynomial- exponential Diophantine equations.