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 an algebraic 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 an open problem. We have recently learned that if the general case could be proved (even a very special version of it), there would be important consequences in determining whether certain upper triangular groups have trivial Poisson boundary, but already the proven case does have implications for this.