Divisibility of integer Laurent polynomials, homoclinic points, and lacunary independence (Q6623999)

This paper uses ideas from algebraic dynamics to find sufficient conditions on Laurent polynomials in \(d\ge1\) variables \(f,p,q\) to ensure that if \(f\) divides \(p+q\) then \(f\) divides \(p\) and \(q\). The (expected) conditions found involve bounds on coefficients and the distance between the supports of \(p\) and \(q\) along with an (unexpected) condition called `atorality' on the complex variety defined by \(f\). The latter condition is related to invertibility of \(f\) when viewed as an element of \(\ell^1(\mathbb{Z}^d)\) and ensures the existence of a non-trivial homoclinic point in an algebraic \(\mathbb{Z}^d\) action which plays a key role in the proofs.