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Dynamical irreducibility of polynomials modulo primes
Virtualconference
Authors : ... (Author of the conference)
... (Publisher )
11L40 - Estimates on character sums
11R09 - Polynomials (irreducibility, etc.)
11R45 - Density theorems
37P25 - Dynamical systems over finite ground fields
... (Publisher )
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Abstract :
In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions, he set of primes $p$ such that $f$ is dynamical irreducible modulo $p$ is of relative density zero. The proof of this result relies on a combination of analytic (the square sieve) and diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe.
Keywords : irreducibility of polynomials; iterates of polynomials
MSC Codes : Keywords : irreducibility of polynomials; iterates of polynomials
11L40 - Estimates on character sums
11R09 - Polynomials (irreducibility, etc.)
11R45 - Density theorems
37P25 - Dynamical systems over finite ground fields
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Information on the Video
Language : EnglishAvailable date : 01/12/2020 Conference Date : 23/11/2020 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Number Theory Format : MP4 (.mp4) - HD Video Time : 00:51:38 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-11-23_Ostafe.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 aléatoireDates : 23/11/2020 - 27/11/2020 Event Year : 2020 Event URL : https://www.chairejeanmorlet.com/2256.html
Citation Data
DOI : 10.24350/CIRM.V.19688303Cite this video as: (2020). Dynamical irreducibility of polynomials modulo primes. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19688303 URI : http://dx.doi.org/10.24350/CIRM.V.19688303 |
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Bibliography
- MÉRAI, László, OSTAFE, Alina, et SHPARLINSKI, Igor E. Dynamical irreducibility of polynomials modulo primes. Mathematische Zeitschrift, 2020, p. 1-13. - https://doi.org/10.1007/s00209-020-02630-5
