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.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/246