Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

Collection Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

Organizer(s) Rivat, Joël ; Tichy, Robert

Date(s) 23/11/2020 - 27/11/2020

linked URL https://www.chairejeanmorlet.com/2256.html

00:00:00 / 00:00:00

9 28

Dynamical irreducibility of polynomials modulo primes

By Alina Ostafe

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.

Information about the video

Date of recording 23/11/2020
Date of publication 01/12/2020
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19688303
Cite this video Ostafe, Alina (23/11/2020). Dynamical irreducibility of polynomials modulo primes. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19688303
URL https://dx.doi.org/10.24350/CIRM.V.19688303

Domain(s)

Number Theory

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

MSC codes

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