Clemens Fuchs's Homepage 
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Clemens Fuchs's Homepage
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Research Interests: I am particularly interested in Number Theory and Algebraic Geometry, especially in the resolution of Diophantine equations, and its applications in theoretical computer science as e.g. cryptography, digital expansions and coding theory. Methods for attacking such problems come from transcendence theory (A. Baker's method in linear forms in logarithms of algebraic numbers), Diophantine approximation (W. Schmidt's Subspace theorem) and other more elementary tools. The problems are studied over the integers, in algebraic number fields, as well as in function fields over fields of characteristic zero. One typical problems that goes back to Diophantus of Alexandria asks for sets of positive integers such that the produt of any two different elements is one less than a perfect square. I have studied many similar questions with A. Dujella. For example we answered a question posed by Euler: There is no set consisting of four positive integers such that the product of any two plus their sum is a perfect square. Other problems I am interested in are Diophantine equations of mixed polynomial-exponential type. Such problems show up by studying arithmetical properties of linear recurring sequences; in particular this is related to deep and important work of P. Corvaja and U. Zannier. Moreover, I have worked on applications of the Brownawell-Masser inequality, e.g. to obtain results on polynomial-exponential equations or on families of Thue equations and inequalities over function fields. I am also interested in dynamical systems obtained by substitutions and their connections to abstract number systems. Research Projects: 2022-2025: Leader (with A. Schröder, W. Schmid and S. Plangg) in the Sparkling-Science project SPA 01-080 MAJA- Mathematische Algorithmen für Jedermann Analysiert funded by BMBWF; for more details see the webpage of the project which you can find here. 2020-2023: Co-Leader (with R.F. Tichy) in the project I4406 - Diophantine number theory funded by the FWF Austrian Science Fund; for more details see the webpage of the project which you can find here. 2014-2018: Co-Leader (with A. Schröder, K.J. Fuchs and W. Schmid) in the Sparkling-Science project SPA 05-172 EMMA - Experimentieren mit mathematischen Algorithmen funded by BMWFW. 2013-2018: Co-Leader (with R.F. Tichy) in the project P26114 - Diophantine Problems: Analytic, geometric and computational aspects funed by the FWF Austrian Science Fund; for more details see the webpage of the project which you can find here. 2012-2016: Leder and Principal investigator in the project P24574 - Explicit Problems in Diphantine Analysis and Geometry funded by the FWF Austrian Science Fund; for more details see the webpage of the project which you can find here. 2009-2012: Co-Leader (with G. Wüstholz and P. Habegger) of the project PDFMP2_122850 - Number Theory. Research Module within the ProDoc "Arithmetic and Geometry" funded by the SNF. 2004-2005: Leader and Principal investigator in the project J2407-N12 - Ineffective and effective methods for Diophantine problems funded by the FWF Austrian Science Fund. |
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