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Audiovisual Mathematics Library

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Virtualconference

Authors : Walsh, Gary (Author of the conference)
CIRM (Publisher )

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Abstract : In a recent paper, Istvan Gaal and Laszlo Remete studied the integer solutions to binary quartic Thue equations of the form $x^4-dy^4 = \pm 1$, and used their results to determine pure quartic number fields which contain a power integral basis. In our talk, we propose a new way to approach this diophantine problem, and we also show how an effective version of the abc conjecture would allow for even further improvements. This is joint work with M.A. Bennett. We also discuss a relation between this quartic diophantine equation to recent joint work with P.-Z. Yuan.

Keywords : index form; integral basis; quadric fields; elliptic curves

MSC Codes :
11D25 - Cubic and quartic equations
11D57 - Multiplicative and norm form equations
11R16 - Cubic and quartic extensions

Information on the Video

Film maker :

Hennenfent, Guillaume

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Download :

https://videos.cirm-math.fr/2020-11-25_Walsh.mp4

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 aléatoire
Event 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.19689803
Cite this video as: Walsh, Gary (2020). On binary quartic Thue equations and related topics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19689803
URI : http://dx.doi.org/10.24350/CIRM.V.19689803

See Also

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Bibliography

GAÁL, István et REMETE, László. Binomial Thue equations and power integral bases in pure quartic fields. arXiv preprint arXiv:1810.00063, 2018. - https://arxiv.org/abs/1810.00063

ZAGIER, Don. Large integral points on elliptic curves. Mathematics of Computation, 1987, vol. 48, no 177, p. 425-436. - http://dx.doi.org/10.1090/S0025-5718-1987-0866125-3

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