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 the study of the orbit length of reduction modulo p of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over the field of complex numbers. Applying results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters.

This is joint work with M.-C. Chang (Riverside), C. D’Andrea (Barcelona), A. Ostafe (New South Wales) and I. Shparlinski (New South Wales).