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arXiv · 2403.20111

Divisibility of Integer Laurent Polynomials, Homoclinic Points, and Lacunary Independence

Douglas Lind·Klaus Schmidt
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Abstract

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.

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Douglas Lind, Klaus Schmidt. 2024-03-29. Divisibility of Integer Laurent Polynomials, Homoclinic Points, and Lacunary Independence. https://arxiv.org/abs/2403.20111

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