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Audiovisual Mathematics Library

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Virtualconference

Authors : Ostafe, Alina (Author of the conference)
CIRM (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 :
11L40 - Estimates on character sums
11R09 - Polynomials (irreducibility, etc.)
11R45 - Density theorems
37P25 - Dynamical systems over finite ground fields

Information on the Video

Film maker :

Hennenfent, Guillaume

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Download :

https://videos.cirm-math.fr/2020-11-23_Ostafe.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.19688303
Cite this video as: Ostafe, Alina (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

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