P24574 - 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 shall*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.

The project is funded by the FWF Austrian Science Fund and has started on 01.09.2012. The place of research was initially at the Institute for Analysis and Computational Number Theory (Math A), TU Graz (Steyrergasse 30/II, 8010 Graz, Austria). The project was shifted to the Department of Mathematics, University of Salzburg (Hellbrunnerstr. 34/I, 5020 Salzburg, Austria) on 01.04.2013. The project has ended on 31.12.2016.

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 shall

The project is funded by the FWF Austrian Science Fund and has started on 01.09.2012. The place of research was initially at the Institute for Analysis and Computational Number Theory (Math A), TU Graz (Steyrergasse 30/II, 8010 Graz, Austria). The project was shifted to the Department of Mathematics, University of Salzburg (Hellbrunnerstr. 34/I, 5020 Salzburg, Austria) on 01.04.2013. The project has ended on 31.12.2016.

Open Positions:
Currently there are no positions open in this project.

Team:

Clemens Fuchs

Christoph Hutle (formerly funded by FWF-P24574)

Christina Karolus

Volker Ziegler

Including:

Fabrizio Barroero (Graz/Pisa/Manchester)

Kwok Chi Chim (Graz; funded by FWF-P26114)

Christian Elsholtz (Graz)

Christopher Frei (Graz/München)

Roswitha Hofer (Linz)

Dijana Kreso (Graz/Salzburg)

Shanta Laishram (Dehli)

Roland Paulin (Bonn; formerly funded by FWF-P24574)

Gulio Peruginelli (Graz; funded by FWF-P23245)

Istvan Pink (Debrecen; formerly funded by FWF-P24801)

Petra Tadic (Pula/Zagreb; formerly funded by FWF-P24574)

Robert Tichy (Graz)

Mingxi Wang (formerly funded by FWF-P24574)

Gisbert Wüstholz (Zurich/Graz; formerly funded by FWF-P26114)

Clemens Fuchs

Christoph Hutle (formerly funded by FWF-P24574)

Christina Karolus

Volker Ziegler

Including:

Fabrizio Barroero (Graz/Pisa/Manchester)

Kwok Chi Chim (Graz; funded by FWF-P26114)

Christian Elsholtz (Graz)

Christopher Frei (Graz/München)

Roswitha Hofer (Linz)

Dijana Kreso (Graz/Salzburg)

Shanta Laishram (Dehli)

Roland Paulin (Bonn; formerly funded by FWF-P24574)

Gulio Peruginelli (Graz; funded by FWF-P23245)

Istvan Pink (Debrecen; formerly funded by FWF-P24801)

Petra Tadic (Pula/Zagreb; formerly funded by FWF-P24574)

Robert Tichy (Graz)

Mingxi Wang (formerly funded by FWF-P24574)

Gisbert Wüstholz (Zurich/Graz; formerly funded by FWF-P26114)

New book: Arithmetic and Geometry: Ten Years in Alpbach (AMS-202) added on November 27, 2019!

PhD defence of C. Karolus: Invitation

PhD thesis of C. Karolus added on September 12, 2019!

arXiv:1810.12141 is accepted for publication in Annales Maths Blaise Pascal!

arXiv:1810.12141 added on October 30, 2018!

arXiv:1703.03258 is accepted for publication in manuscripta mathematica!

arXiv:1711.04668 is accepted for publication in Res. Number Theory!

PhD defence of C. Hutle: Invitation

PhD thesis of C. Hutle added on February 28, 2018!

arXiv:1711.04668 added on November 15, 2017!

arXiv:1705.05044 added on May 17, 2017!

G. Wüstholz, C. Fuchs: Arithmetic and Geometry: Ten Years in Alpbach, Annals of Mathematics Studies 202, PUP, Princeton, NJ, 2019

C. Karolus, Composition of polynomials, lacunarity, and linear recurrences, PhD thesis, University of Salzburg, 2019

C. Fuchs, C. Karolus, Composite values of polynomial power sums, arXiv:1810.12141, Annales Maths Blaise Pascal 26 (2019), 1-24

C. Hutle, Diophantine triples and linear recurrence sequences, PhD thesis, University of Salzburg, 2018

D. Kreso, Diophantine equations in separated variables and lacunary polynomials, arXiv:1705.05044, Int. J. Number Theory 13 (2017), 2055-2074

C. Fuchs, C. Karolus, D. Kreso, Decomposable polynomials in second order linear recurrence sequences, arXiv:1703.03258, manuscripta mathematica 159(3) (2019), 321-346

C. Karolus, Explicit bounds for composite lacunary polynomials, arXiv:1704.04292, Glas. Mat. Ser. III 54(74) (2019), 11-20

R. Paulin, On the minimal degree of morphisms between algebraic curves, arXiv:1608.08319

C. Fuchs, C. Hutle and F. Luca, Diophantine triples in linear recurrence sequences of Pisot type, arXiv:1711.04668, Res. Number Theory 4 (2018), 4:29

C. Fuchs, C. Hutle, F. Luca, L. Szalay, Diophantine triples and k-generalized Fibonacci sequences, arXiv:1602.08236, Bulletin of the Malaysian Mathematical Sciences Society 41(3) (2018), 1449-1465, PDF

C. Fuchs, V. Mantova and U. Zannier, On fewnomials, integral points and a toric version of Bertini's theorem, arXiv:1412.4548, JAMS 31 (2018), 107-134

C. Fuchs, C. Hutle, N. Irmak, F. Luca, L. Szalay, Only finitely many Tribonacci Diophantine triples exist, arXiv:1508.07760, Math. Slovaca 67 (2017), 853-862

C. Fuchs, L. Hajdu, 30 years of collaboration, arXiv:1511.07689, Period. Math. Hung. 74 (2017), 255-274

M. Schreyer, R. Paulin and W. Trutschnig, On the exact region determined by Kendall's tau and Spearman's rho, arXiv:1502.04620, J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017), 613-633

R. Paulin, Largest values of the Stern sequence, alternating binary expansions and continuants, arXiv:1608.08322, J. Integer Seq. 20 (2017) Article 17.2.8

M. Wang, A dynamical Mordell Lang property on the disc, arXiv:1403.3975, Trans. Amer. Math. Soc. 369 (2017), 2183-2204

C. Fuchs and H.D. Pham, Some applications of the p-adic analytic subgroup theorem, arXiv:1412.1248, Glas. Mat. 51(71) (2016), 335-343

L. Hajdu, S. Laishram, M. Szikszai, Perfect powers in products of terms of elliptic divisibility sequences, arXiv:1604.03707, Bull. Austr. Math. Soc. 94 (2016), 395-404

C. Fuchs, A. Jurasic and R. Paulin, Elementary resolution of a family of quartic Thue equations over function fields, arXiv:1412.3216, Monatsh. Math. 180 (2016), 205-211

R. Paulin, An explicit Andre-Oort type result for P^1(C)xG_m(C) based on logarithmic forms, arXiv:1403.2949, Publ. Math. Debrecen 88 (2016), 21-33

R. Paulin, An explicit Andre-Oort type result for P^1(C)xG_m(C), arXiv:1403.2264, Math. Proc. Cambridge Philos. Soc. 159 (2015), 153-163

Q. Lin and M. Wang, Isogeny orbits in a family of abelian varieties, arXiv:1403.3976, Acta Arith. 170 (2015), 161-173

C. Fuchs and H.D. Pham, Commutative algebraic groups and p-adic linear forms, arXiv:1404.3975, Acta Arith. 169 (2015), 115-147

I. Gusic and P. Tadic, Injectivity of the specialization homorphism of elliptic curves, arXiv:1211.3851, J. Number Theory 148 (2015), 137-152

C. Fuchs and H.D. Pham, The p-adic analytic subgroup theorem revisited, arXiv:1502.00768, p-Adic Number, Ultrametric Analysis and Applications 7 (2015), 143-156

A. Dujella and C. Elsholtz, Sumsets being squares, Acta Math. Hungar. 141 (2013), 353-357

Y.F. Bilu, C. Fuchs, F. Luca and A. Pinter, Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations, Publ. Math. Debrecen 82/1 (2013), 219-254

P. Tadic, The rank of certain subfamilies of the elliptic curve y^2=x^3-X+T^2, Ann. Math. Inform. 40 (2012), 145-153

C. Fuchs and A. Dujella, On a problem of Diophantus for rationals, J. Number Theory 132 (2012), 2075-2083

C. Fuchs and U. Zannier, Composite rational functions expressible with few terms, J. Eur. Math. Soc. (JEMS) 14 (2012), 175-208

Attile Pethö (University of Debrecen), 09.-12.09.2019

Andrej Dujella (Zagreb), 26.-27.02.2018

Jan-Hendrik Evertse (Leiden), 03.-10.04.2016

Gisbert Wüstholz (Zürich/Graz), 14.-20.02.2016

Martin Widmer (London), 14.-20.02.2016

Michael Stoll (Bayreuth), 14.-20.02.2016

Kalman Györy (Debrecen), 14.-20.02.2016

Andrej Dujella (Zagreb), 14.-20.02.2016

Yui Bilu (Bordeauy), 14.-20.02.2016

Michael Bennett (Vancouver), 14.-20.02.2016

Shanta Laishram (Dehli), 19.-25.04.2015

Florian Luca (Mexico), 21.-26.04.2015

Yann Bugeaud (Strasbourg), 28.01.-02.02.2015

Kwok Chi Chim (Graz), 03.-06.12.2014

Gisbert Wüstholz (Zurich/Graz), 03.-06.12.2014

Vincenzo Mantova (Camerino/Pisa), 12.-14.11.2014

Lajos Hadju (University of Debrecen), 25.-30.11.2013

Attila Pethö (University of Debrecen), 25.-30.11.2013

Akos Pinter (University of Debrecen), 25.-30.11.2013

Rafael von Känel (MPI Bonn), 25.-30.11.2013

Andrej Dujella (University of Zagreb), 25.-30.11.2013

Francesco Veneziano (TU Graz/ESI), 14.-16.11.2013

Gisbert Wüstholz (ETH Zurich), 12.-13.11.2013

Fabrizio Barroero (TU Graz), 17.10.2013

Mingxi Wang (TU Darmstadt), 20.-21.06.2013

Hiep Pham (ETH Zurich), 23.-24.05.2013

Christian Elsholtz (TU Graz), 15.-16.05.2013

Akos Pinter (University of Debrecen), 02.04.2013

Andrej Dujella (University of Zagreb), 15.-16.11.2012

Tomislav Pejkovic (University of Zagreb), 15.-16.11.2012

Vinko Petricevic (University of Zagreb), 15.-16.11.2012

Attila Pethö (University of Debrecen), 07.-11.11.2012

ELAZ2016, Strobl/Austria, 05.-09.09.2016

Workshop "Arithmetic and Geometry: Ten years in Alpbach", Alpbach/Austria, 26.06.-01.07.2016

14th Meeting of the Canadian Number Theory Association (CNTA XIV), Calgary/Canada, 20.06.-24.06.2016

Current Trends in Diophantine Geometry and Transcendence, Taipei/Taiwan, 23.-27.05.2016

Diophantische Approximationen, Oberwolfach/Germany, 10.-16.04.2016

Computational aspects of Diophantine equations, Salzburg/Austria, 15.-19.02.2016

Graz-Salzburg Workshop on Diophantine Problems, Graz/Austria, 26.-27.11.2015

Joint Austrian-Hungarian Mathematical Conference 2015, Györ/Hungary, 25.-27.08.2015

29th Journees Arithmetiques (JA2015), Debrecen/Hungary, 06.-10.07.2015

Workshop "Special cycles on Shimura curves", Alpbach/Austria, 05.-10.07.2015

Final ERC Meeting in Diophantine Geometry, Rome/Italy, 27.-28.05.2015

Pre-Alpbach Minicourse - Special cycles on Shimura curves, Zurich/Switzerland, 30.03.-01.04.2015

Conference on Number Theory and Algebra (on the occasion of the 60th birthday of Ivica Gusic), Zagreb/Croatia, 26.-28.11.2014

Conference on Diophantine m-tuples and related problems, Westville/USA, 13.-15.11.2014

CIRM-Conference "Diophantine approximation and transcendence", Luminy/France, 15.-19.09.2014

ERC-conference ``Diophantine Geometry, Unlikely Intersections and Algebraic Dynamics'', Cetraro/Italy, 14.-19.07.2014

Workshop "Periods and heights of CM abelian varieties", Alpbach/Austria, 06.-11.07.2014

Graz-Salzburg Workshop on Diophantine Problems, Graz/Austria, 19.-20.05.2014

Preparation for Alpbach 2014 (by V. Maillot and Damian Roessler), Zurich/Switzerland, 07.-10.04.2014

AAA87 - 87th Workshop on General Algebra, Linz/Austria, 07.-09.02.2014

CIRM-Conference "Unlikely Intersections", Luminy/France, 03.-07.02.2014

ESI-Miniworkshop on ``Explicit Problems in Diophantine Analysis and Geometry'', Vienna/Austria, 29.-30.11.2013

ESI-Workshop, Vienna/Austria, 25.-29.11.2013

ESI-Program "Heights in Diophantine geometry, group theory and additive combinatorics", Vienna/Austria, 21.10.-20.12.2013

W.M. Schmidt 80, Vienna/Austria, 09.-11.10.2013

Thue 150, Bordeaux/France, 30.09.-04.10.2013

18th ÖMG Congress and Annual DMV Meeting, Innsbruck/Austria, 23.-27.09.2013

Frits60 - "Special functions and special numbers". A conference in honour of Frits Beukers' 60th birthday, Utrecht/The Netherlands, 10.-12.07.2013

ProDoc-Workshop "p-adic Modular Forms", Alpbach/Austria, 07.-12.07.2013

28th Journées arithmétiques 2013, Grenoble/France, 01.-05.07.2013

25 years of Number Theory Seminars, FIM at ETH Zurich, Zurich/Switzerland, 03.-08.06.2013

BIRS-Workshop 13w5141 "The art of iterating rational functions over finite fields", Banff/Canada, 05.-10.05.2013

Research visit at ETH Zurich, Zurich/Switzerland, 17.-19.02.2013

ERC Intensive Research Period in Diophantine Geometry, Pisa/Italy, 20.-27.09.2012

ProDoc-Workshop "Multiple-Zeta-Values", Alpbach/Austria, 02.-07.09.2012

Impressum 17.02.2020