# -- CANCELLED -- Stability of polynomials modulo primes

Posted in
Speaker:
Laszlo Merai
Affiliation:
RICAM (Linz, Austria)
Date:
Wed, 2020-04-29 14:30 - 15:30
Location:
MPIM Lecture Hall
Parent event:
Number theory lunch seminar

For a polynomial $f\in\mathbb{K}[X]$ over some field $\mathbb{K}$ we define the sequence of polynomials
$$f^0(X)=X, \quad \text{and} \quad f^{(n)}(X)=f(f^{(n-1)}(X)), \quad n=1,2,\dots$$
The polynomial $f$ is said to be stable if all iterates $f^{(n)}$ are irreducible.

It is conjectured, that for a quadratic polynomial $f\in\mathbb{Z}[X]$, its reduction $f_p\in\mathbb{F}_p[X]$ modulo $p$ can be stable just for finitely many primes $p$.

In this talk I show that a weakened version of this conjecture is true, that is, the reduction $f_p$ can be stable for a set of primes $p$ of relative density zero.

This is a joint work with Alina Ostafe and Igor E. Shparlinski.

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