The mathematical study of Diophantine problems, that are problems to be solved in integers, rationals and their generalizations to number fields and function fields, is called Diophantine Analysis. Main tools that are used for solving such problems include the Thue-Siegel-Roth-Schmidt method and Baker's method on linear forms in logarithms of algebraic numbers for number fields and the generalization of the ABC-theorem due to Brownawell and Masser for function fields. Since working with equations is somewhat limited, it is a modern approach to study Diophantine problems that have also a geometrical meaning; this is the approach which is studied in Diophantine Geometry. Here the solution set to a Diophantine problem is viewed e.g. as a scheme of finite type over the spectrum of the ring of integers and then also tools from algebraic geometry are used to study it.
In this project we shall investigate several further and new explicit Diophantine problems
. We shall often take a more geometric point of view. These questions include problems on lacunary rational functions that are composite, rational and integral points on fibered surfaces also in connection with problems related to linear recurrences and families of classical Diophantine equations over function fields, and Diophantine tuples.
The project is funded by the FWF Austrian Science Fund
and has started on 01.09.2012. The place of research was initially at the Institute for Analysis and Computational Number Theory (Math A)
, TU Graz
(Steyrergasse 30/II, 8010 Graz, Austria). The project was shifted to the Department of Mathematics
, University of Salzburg
(Hellbrunnerstr. 34/I, 5020 Salzburg, Austria) on 01.04.2013. The project has ended on 31.12.2016.