  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.
