On some algebraic and arithmetic aspects in polynomial dynamics

Abstract:

Algebraic Dynamical Systems over finite fields, that is, dynamical systems generated by iterations of polynomials and rational functions, are exciting and challenging mathematical objects with intricate algebraic and number theoretic properties and very complex behaviour. Controlling algebraic properties of iterates is a fundamental mathematical problem whose solution has multiple implications, but little progress has been made in this direction.

In this talk I will outline some algebraic and number theoretic properties of such dynamical systems which include: degree growth, irreducibility and irreducible divisors of polynomial iterates, and use explicit versions of Hilbert's Nullstellensatz to obtain results about periodic points and intersection of orbits in reductions modulo primes of polynomial dynamical systems.