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

TL;DR

This paper establishes bounds on primes for which reductions of polynomial systems over integers behave differently over finite fields, and applies these results to study periodic points and orbit intersections in algebraic dynamical systems, linking to the dynamical Mordell-Lang conjecture.

Contribution

It provides new bounds for primes affecting polynomial system reductions and connects these bounds to properties of algebraic dynamical systems over finite fields.

Findings

Bounds on primes for polynomial system reductions

Analysis of periodic points in algebraic dynamical systems

Connections to the uniform dynamical Mordell-Lang conjecture

Abstract

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.