Divisibility of integer Laurent polynomials and dynamical systems

Let f, p, and q be Laurent polynomials in one or several variables with integer coefficients, and suppose that f divides p + q. In joint work with Klaus Schmidt, 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 f called atorality about how the complex variety of f intersects the multiplicative unit torus. The proof uses a dynamical system related to f and the fundamental dynamical notion of homoclinic point. Without the atorality assumption this method fails, the validity of our result in this case remains open problem.