CIRM - Videos & books Library - Skolem's conjecture and exponential Diophantine equations

Virtualconference

Authors : Hajdu, Lajos (Author of the conference)
CIRM (Publisher )

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Abstract : Exponential Diophantine equations, say of the form (1) $u_{1}+...+u_{k}=b$ where the $u_{i}$ are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the solutions only in case of k = 2 (by results of Gyory and others, based upon Baker's method), for k > 2 only the number of so-called non-degenerate solutions can be bounded (by the Thue-Siegel-Roth-Schmidt method; see also results of Evertse and others). In particular, there is a big need for a method which is capable to solve (1) completely in concrete cases.
Skolem's conjecture (roughly) says that if (1) has no solutions, then it has no solutions modulo m with some m. In the talk we present a new method which relies on the principle behind the conjecture, and which (at least in principle) is capable to solve equations of type (1), for any value of k. We give several applications, as well. Then we provide results towards the solution of Skolem's conjecture. First we show that in certain sense it is 'almost always' valid. Then we provide a proof for the conjecture in some cases with k = 2, 3. (The handled cases include Catalan's equation and Fermat's equation, too - the precise connection will be explained in the talk). Note that previously Skolem's conjecture was proved only for k = 1, by Schinzel.
The new results presented are (partly) joint with Bertok, Berczes, Luca, Tijdeman.

Keywords : exponential Diophantine equations; Skolem's conjecture

MSC Codes :
11D41 - "Higher degree equations; Fermat's equation"
11D61 - Exponential equations
11D79 - Congruences in many variables

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2020-11-24_Hajdu.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.19687703
Cite this video as: Hajdu, Lajos (2020). Skolem's conjecture and exponential Diophantine equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19687703
URI : http://dx.doi.org/10.24350/CIRM.V.19687703

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Bibliography

HAJDU, L. et TIJDEMAN, R. Skolem's conjecture confirmed for a family of exponential equations. Acta Arithmetica, 2020, vol. 192, p. 105-110. - http://dx.doi.org/10.4064/aa190114-25-2

BERTÓK, Csanád et HAJDU, Lajos. A Hasse-type principle for exponential Diophantine equations and its applications. Mathematics of Computation, 2016, vol. 85, no 298, p. 849-860. - http://dx.doi.org/10.1090/mcom/3002

BERTÓK, Csanád et HAJDU, Lajos. A Hasse-type principle for exponential Diophantine equations over number fields and its applications. Monatshefte für Mathematik, 2018, vol. 187, no 3, p. 425-436. - https://doi.org/10.1007/s00605-018-1169-8

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