Divisibility of Integer Laurent Polynomials, Homoclinic Points, and Lacunary Independence

Let f, p, and q be Laurent polynomials with integer coefficients in one or several variables, and suppose that f divides $p+q$. We establish sufficient conditions to guarantee that f individually divides p and q. These conditions involve a bound on coefficients, a separation between the supports of p and q, and, surprisingly, a requirement on the complex variety of f called atorality satisfied by many but not all polynomials. Our proof involves a related dynamical system and the fundamental dynamical notion of homoclinic point. Without the atorality assumption our methods fail, and it is unknown whether our results hold without this assumption.