Modular Reduction of Systems of Polynomial Equations and Algebraic Dynamical Systems

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell--Lang conjecture.