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

Reductions Modulo Primes of Systems of Polynomial Equations and Algebraic Dynamical Systems