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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2402.13712 |
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| _version_ | 1866929250844540928 |
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| author | Young, Marley |
| author_facet | Young, Marley |
| contents | We classify the pairs of polynomials $f,g \in \mathbb{C}[X]$ having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given $f_1,\ldots, f_n$ from a certain class of polynomials with integer coefficients, the vectors of indices $(m_1,\ldots,m_n)$ such that $f_1^{m_1}(0),\ldots,f_n^{m_n}(0)$ are multiplictively dependent are sparse. We also classify the pairs $f,g \in \mathbb{Q}[X]$ such that there are infinitely many $(x,y) \in \mathbb{Z}^2$ satisfying $f(x)^k=g(y)^\ell$ for some (possibly varying) non-zero integers $k,\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13712 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On multiplicative dependence between elements of polynomial orbits Young, Marley Number Theory Dynamical Systems 37F10, 37P15, 11N25, 11D41 We classify the pairs of polynomials $f,g \in \mathbb{C}[X]$ having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given $f_1,\ldots, f_n$ from a certain class of polynomials with integer coefficients, the vectors of indices $(m_1,\ldots,m_n)$ such that $f_1^{m_1}(0),\ldots,f_n^{m_n}(0)$ are multiplictively dependent are sparse. We also classify the pairs $f,g \in \mathbb{Q}[X]$ such that there are infinitely many $(x,y) \in \mathbb{Z}^2$ satisfying $f(x)^k=g(y)^\ell$ for some (possibly varying) non-zero integers $k,\ell$. |
| title | On multiplicative dependence between elements of polynomial orbits |
| topic | Number Theory Dynamical Systems 37F10, 37P15, 11N25, 11D41 |
| url | https://arxiv.org/abs/2402.13712 |