Divisibility of Integer Laurent Polynomials, Homoclinic Points, and Lacunary Independence
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916204228116480 |
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| author | Lind, Douglas Schmidt, Klaus |
| author_facet | Lind, Douglas Schmidt, Klaus |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20111 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisibility of Integer Laurent Polynomials, Homoclinic Points, and Lacunary Independence Lind, Douglas Schmidt, Klaus Dynamical Systems Primary: 37A15, 13F20, 37A44, Secondary: 37B40, 13F20 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. |
| title | Divisibility of Integer Laurent Polynomials, Homoclinic Points, and Lacunary Independence |
| topic | Dynamical Systems Primary: 37A15, 13F20, 37A44, Secondary: 37B40, 13F20 |
| url | https://arxiv.org/abs/2403.20111 |