Abstract:
Let f, p, and q be Laurent polynomials with integer coefficients in one or several variables, and suppose that f divides p+q. I'll discuss sufficient conditions to guarantee that f individually divides p and q. These conditions involve a bound on coefficients of p and q, a separation between the supports of p and q, and, surprisingly, a requirement on the complex variety of f called atorality.
The proof involves a related dynamical system and the notion of summable homoclinic points of that system.
This polynomial divisibility result has a number of consequences concerning exponential recurrence and the finitary Bernoulli property of the related dynamical system. I'll conclude with some extensions and open problems.
This is joint work with Doug Lind.