I4406 - Diophantine Number Theory

In this joint project of the FWF and the NKFIH we shall use different techniques from Diophantine number theory (as the subspace theorem, the theory of linear forms in logarithms, Runge's method, hypergeometric methods, etc.) to study various Diophantine problems. E.g. we work on Ritt's decomposition theory and Diophantine applications, we shall study some classical Diophantine equations (as the ErdÃ¶s-Straus conjecture, Goormaghtigh's equation and the generalized Ramanujan-Nagell equation, Thue and relative Thue equations), and we shall investigate Diphantine problems with recurrence sequences. We emphasize that the Austrian and Hungarian research groups have a long standing and very fruitful cooperation, which is exceptional even by the international standards. Through this project we want to maintain this tight scientific bond and in particular encourage collaboration between younger members of the groups.

In this joint project of the FWF and the NKFIH we shall use different techniques from Diophantine number theory (as the subspace theorem, the theory of linear forms in logarithms, Runge's method, hypergeometric methods, etc.) to study various Diophantine problems. E.g. we work on Ritt's decomposition theory and Diophantine applications, we shall study some classical Diophantine equations (as the ErdÃ¶s-Straus conjecture, Goormaghtigh's equation and the generalized Ramanujan-Nagell equation, Thue and relative Thue equations), and we shall investigate Diphantine problems with recurrence sequences. We emphasize that the Austrian and Hungarian research groups have a long standing and very fruitful cooperation, which is exceptional even by the international standards. Through this project we want to maintain this tight scientific bond and in particular encourage collaboration between younger members of the groups.

The project has started on 01.04.2020.

Team:

Clemens Fuchs (Salzburg; co-PI)

Kalman Gyõry (Debrecen; PI)

Lajos Hajdu (Debrecen; co-PI)

Sebastian Heintze (Salzburg)

Robert Tichy (Graz; PI)

Ingrid Vukusic (Salzburg; funded by FWF-I4406)

Volker Ziegler (Salzburg)

Clemens Fuchs (Salzburg; co-PI)

Kalman Gyõry (Debrecen; PI)

Lajos Hajdu (Debrecen; co-PI)

Sebastian Heintze (Salzburg)

Robert Tichy (Graz; PI)

Ingrid Vukusic (Salzburg; funded by FWF-I4406)

Volker Ziegler (Salzburg)

In order to foster cooperation in times of Covid-19 we start with a reserach seminar in which results related to the project are presented and discussed. All team members on the Austrian and Hungarian side are cordially invited to participate; link and password of the browser based and self-explanatory Webex meetings will be sent by email. In case you are interested to attend in one of the talks, please send an email to clemens.fuchs AT sbg.ac.at.

Time: Friday, 2.00-3.15pm

Place: Online

Upcoming talks:

- Sebastian Heintze (Salzburg): On two variants of Diophantine tuples, Friday, 9 October 2020
- Abstract: We will consider two different variants of Diophantine tuples occurring in recent papers. The first one is an S-unit variant in number fields and the second one a polynomial variant using linear recurrences. In both cases we will sketch the proof given in the respective paper.

- Istvan Pink (Debrecen): TBA, Friday, 16 October 2020

TBA

C. Fuchs, S. Heintze, Diophantine equations in separated variables and polynomial power sums, arXiv:2008.10342

C. Fuchs, S. Heintze, A function field variant of Pillai's problem, arXiv:2008.10339

C. Fuchs, S. Heintze, Integral zeros of a polynomial with linear recurrences as coefficients, arXiv:2008.10328

R. Tichy, I. Vukusic, D. Yang, V. Ziegler, Integers representable as differences of linear recurrence sequences, arXiv:2008.00844

C. Fuchs, S. Heintze: A polynomial variant of Diophantine triples in linear recurrences, arXiv:2006.12173

C. Fuchs, S. Heintze: Norm form equations with solutions taking values in a multi-recurrence, arXiv:2006.11075

C. Fuchs, S. Heintze: On the growth of linear recurrences in function fields, Bull. Austr. Math. Soc., to appear, arXiv:2006.11074

C. Fuchs, S. Heintze: Another S-unit variant of Diophantine tuples, PAMS, to appear, arXiv:1910.09285

TBA

TBA

Impressum 25.09.2020