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Effective finiteness results for diophantine equations over finitely generated domains
Virtualconference
Authors : Györy, Kalman (Author of the conference)
CIRM (Publisher )
11D57 - Multiplicative and norm form equations
11D61 - Exponential equations
11D72 - Equations in many variables, See also {11P55}
CIRM (Publisher )
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Abstract :
In the 1980's we developed an effective specialization method and used it to prove effective finiteness theorems for Thue equations, decomposable form equations and discriminant equations over a restricted class of finitely generated domains (FGD's) over $\mathbb{Z}$ which may contain not only algebraic but also transcendental elements. In 2013 we refined with Evertse the method and combined it with an effective result of Aschenbrenner (2004) concerning ideal membership in polynomial rings over $\mathbb{Z}$ to establish effective results over arbitrary FGD's over $\mathbb{Z}$. By means of our method general effective finiteness theorems have been obtained in quantitative form for several classical Diophantine equations over arbitrary FGD's, including unit equations, discriminant equations (Evertse and Gyory, 2013, 2017), Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation (Bérczes, Evertse and Gyory, 2014), generalized unit equations (Bérczes, 2015), and the Catalan equation (Koymans, 2015). In the first part of the talk we shall briefly survey these results. Recently we proved with Evertse effective finiteness theorems in quantitative form for norm form equations, discriminant form equations and more generally for decomposable form equations over arbitrary FGD's. In the second part, these new results will be presented. Some applications will also be discussed.
Keywords : Diophantine equations; effective results over FGD's
MSC Codes : Keywords : Diophantine equations; effective results over FGD's
11D57 - Multiplicative and norm form equations
11D61 - Exponential equations
11D72 - Equations in many variables, See also {11P55}
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 01/12/2020 Conference Date : 24/11/2020 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Number Theory Format : MP4 (.mp4) - HD Video Time : 00:47:07 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-11-24_Giory.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoireEvent Organizers : Rivat, Joël ; Tichy, Robert Dates : 23/11/2020 - 27/11/2020 Event Year : 2020 Event URL : https://www.chairejeanmorlet.com/2256.html
Citation Data
DOI : 10.24350/CIRM.V.19687403Cite this video as: Györy, Kalman (2020). Effective finiteness results for diophantine equations over finitely generated domains. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19687403 URI : http://dx.doi.org/10.24350/CIRM.V.19687403 |
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Bibliography
- EVERTSE, J.H.; GYORY, K; Effective results and methods for Diophantine equations over finitely generated methods, Book, submited for publication -
