Ineffective and Effective Methods for Diophantine Problems

This project is devoted to the resolutions of Diophantine Problems, which are problems to be solved in integers and not for instance in real numbers. To achieve this several different methods are applied, namely so called ineffective and effective methods. In the year 1970 Matijasevic gave a negative answer to Hilbert`s 10th problem, which asks for a universal algorithm for the decidability of a Diophantine equation. Since this time we are concerned with the question to find classes of Diophantine problems (also inequalities and other problems are of interest), as large as possible, for which we can prove the algorithmic decidability. Also weaker questions are of interest, namely the qualitative question whether or not a class of problems has only finitely many solutions or the problem of calculating upper bounds for the number of solutions depending on certain parameters of the class, such results are called quantitative results. These questions are the main topic of this project and several different methods are used to handle them. Ineffective methods as the well-know Subspace Theorem, which goes back to the Austrian mathematician W.M. Schmidt, allow to obtain qualititative and quantitative results, whereas effective methods as Baker`s method of linear forms in logarithms of algebraic numbers, enable us to get algorithmic results. Several concrete problems are mentioned in the proposal. The first part is mainly devoted to Diophantine equations and inequalities, particularly where linear recurring sequences are involved. In the second part new constructions for elliptic curves are investigated, which were recently introduced by A. Dujella. These constructions have applications in cryptography.