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. We use this to establish exponential recurrence of the related dynamical system, and conclude with some remarks and open problems.