Explicit Problems in Diophantine Analysis and Geometry

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 suggest to 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. We highlight two specific questions that are posed. In one of the projects we try to characterize composite rational functions that are lacunary. Composite rational functions are of importance in many areas, e.g. in dynamics or in the study of Diophantine equations with separate variables. Also lacunary rational functions appear in many contexts, we mention problems coming from real geometry or functions on tori. The aim is to prove that the composition factors of a composite lacunary rational function can be described by effectively computable data unless the inner composition factor has a very special exceptional shape that can be explicitly given and in which case the decomposition is trivial. Related questions are also posed e.g. to give a geometric reformulation of this problem or concerning variants of the notion of lacunarity. One of the main tools in the proofs will be the Brownawell-Masser theorem that was mentioned above. This will also be a main tool in many of the other questions which are concerned with problems related to polynomials. Another main project is concerned with so-called Diophantine tuples that are sets consisting of positive integers such that the product of any two increased by one is a square of an integer. This and related notions got a lot of attention in the past since Diophantine tuples were already studied by Diophantus and later investigated by Fermat, Euler, Baker, Dujella and others. One of the many conjectures in this subject says that there is no such sets consisting of five elements with this property. We shall try to make progress on this conjecture as well as on related questions that ask comparable things when the elements are polynomials instead of integers or the product of any two is an element in a given linear recurrence instead of being a square.