Primitive prime divisors in the forward orbit of a polynomial
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| Format: | Preprint |
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2025
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| _version_ | 1866929698688204800 |
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| author | Laishram, Shanta Rout, Sudhansu S. Yadav, Prabhakar |
| author_facet | Laishram, Shanta Rout, Sudhansu S. Yadav, Prabhakar |
| contents | For the polynomial $f(z) \in \mathbb{Q}[z]$, we consider the Zsigmondy set $\mathcal{Z}(f,0)$ associated to the numerators of the sequence $\{f^n(0)\}_{n \geq 0}$. In this paper, we provide an upper bound on the largest element of $\mathcal{Z}(f, 0)$. As an application, we show that the largest element of the set $\mathcal{Z}(f,0)$ is bounded above by $6$ when $f(z) = z^d + z^e +c \in \mathbb{Q}[z]$, with $d>e \geq 2$ and $|c|>2$. Furthermore, when $f(z) =z^d+c \in \mathbb{Q}[z]$ with $|f(0)| > 2^{\frac{d}{d-1}}$ and $d >2$, we also deduce a result of Krieger [Int. Math. Res. Not. IMRN, 23 (2013), pp. 5498-5525] as a consequence of our main result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02600 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primitive prime divisors in the forward orbit of a polynomial Laishram, Shanta Rout, Sudhansu S. Yadav, Prabhakar Number Theory Dynamical Systems 11B37 (Primary), 37F10, 37P05 (Secondary) For the polynomial $f(z) \in \mathbb{Q}[z]$, we consider the Zsigmondy set $\mathcal{Z}(f,0)$ associated to the numerators of the sequence $\{f^n(0)\}_{n \geq 0}$. In this paper, we provide an upper bound on the largest element of $\mathcal{Z}(f, 0)$. As an application, we show that the largest element of the set $\mathcal{Z}(f,0)$ is bounded above by $6$ when $f(z) = z^d + z^e +c \in \mathbb{Q}[z]$, with $d>e \geq 2$ and $|c|>2$. Furthermore, when $f(z) =z^d+c \in \mathbb{Q}[z]$ with $|f(0)| > 2^{\frac{d}{d-1}}$ and $d >2$, we also deduce a result of Krieger [Int. Math. Res. Not. IMRN, 23 (2013), pp. 5498-5525] as a consequence of our main result. |
| title | Primitive prime divisors in the forward orbit of a polynomial |
| topic | Number Theory Dynamical Systems 11B37 (Primary), 37F10, 37P05 (Secondary) |
| url | https://arxiv.org/abs/2502.02600 |