  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.
