CIRM - Videos & books Library - Effective finiteness results for diophantine equations over finitely generated domains

Virtualconference

Auteurs : Györy, Kalman (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : 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

Codes MSC :
11D57 - Multiplicative and norm form equations
11D61 - Exponential equations
11D72 - Equations in many variables, See also {11P55}

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2020-11-24_Giory.mp4

Outreach

Nom de la rencontre : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
Organisateurs de la rencontre : Rivat, Joël ; Tichy, Robert
Dates : 23/11/2020 - 27/11/2020
Année de la rencontre : 2020
URL Congrès : https://www.chairejeanmorlet.com/2256.html

Données de citation

DOI : 10.24350/CIRM.V.19687403
Citer cette vidéo: 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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Bibliographie

EVERTSE, J.H.; GYORY, K; Effective results and methods for Diophantine equations over finitely generated methods, Book, submited for publication -

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