S-Unit equations and their applications

The concern of this project is the effective solution of S-unit equations in view of applications. A (weighted) S- Unit equation is a linear equation over a finitely multiplicative subgroup G of K, where K is usually a number field and G is the group of S-units. S-unit equations have been studied for their own interest and also in view of applications. These investigations were initiated by Siegel and Mahler who proved the finiteness of rational and or integral points on certain curves. Lang was the first who noted the importance of studying S-unit equations as an own object. For S-Unit equations in three variables it is possible to solve effectively S-unit equations by applying Baker`s method, but for more than three variables (up to now) one has to apply ineffective methods such as Schmidt`s subspace theorem and its generalizations. In this project we do not restrict ourselves to S-unit equations (and their applications) in number fields, but we are also interested in applications in global function fields. In contrast to the number field case effective methods are known. As mentioned above S-unit equations may be applied to various problems. For instance norm form equations can be solved by using the theory of S-unit equations. Such norm form equations were studied by several authors. Within this project the proposer plans to investigate arithmetic progressions in the set of solutions of such norm form equations. Arranging the solutions in an n times H array, where n is the number of unknowns and H is the number of solutions, we distinguish between the horizontal (finding solutions that are in arithmetic progression) and the vertical problem (finding solutions for which the i-th components are in arithmetic progression). In this project the proposer plans to investigate both problems. A special case of norm form equations are so called Thue equations. In particular binary Thue equations are planed to be studied. In recent time such Thue equations were successfully tackled by a combination of various methods, as Frey curve techniques, theory of cyclotomic fields, linear forms in logarithms etc. The proposer plans to continue and to generalize these investigations. In particular, binary Thue-Mahler equations and relative binary Thue equations will be considered. Recently Jarden and Narkiewicz found a further application of S-unit equations, when they showed that there does not exist a number k such that each algebraic integer of a fixed number field can be written as the sum of exactly k units. Moreover, they showed that this is also true for every finitely generated integral domain, i.e. the non finiteness of the unit sum number (for a precise definition see the proposal). However, the question remains which number fields have the property that every integer can be written as a sum of units. The quadratic case was settled by Belcher, Ashrafi and Vamos and the case of complex cubic, complex purely quartic and complex bi-quadratic fields was solved by the proposer in cooperation with Filipin and Tichy. Similar investigations led them to an asymptotic formula for the number of non-associated algebraic integers that can be written as the sum of exactly k units in real quadratic number fields. Such results were obtained by utilizing the theory of S-unit equations and the proposer plans to continue such investigations. Moreover, several generalizations of the unit sum number problem will be considered. It is planed to accomplish this project at the university of Debrecen in cooperation with the number theory group of Kalman Györy. His group consists of leading experts in the field of S-unit equations and so I will have the best possible conditions to work successfully on this project.