Prime-powered images and irreducible polynomials in dynamical semigroups
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| Format: | Preprint |
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2025
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| _version_ | 1866911204228726784 |
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| author | Bhardwaj, Aristaa Boyer-Paulet, Adrian Hindes, Wade Qiu, Emma Sun, Alexander |
| author_facet | Bhardwaj, Aristaa Boyer-Paulet, Adrian Hindes, Wade Qiu, Emma Sun, Alexander |
| contents | Let $G=\langle x^d+c_1,\dots,x^d+c_s\rangle$ be a semigroup generated under composition for some $c_1,\dots,c_s\in\mathbb{Z}$ and some $d\geq2$. Then we prove that, outside of an exceptional one-parameter family, $G$ contains a large and explicit subset of irreducible polynomials if and only if it contains at least one irreducible polynomial. In particular, this conclusion holds when $G$ is generated by at least $s\geq3$ polynomials when $d$ is odd and at least $s\geq5$ polynomials when $d$ is even. To do this, we prove a classification result for prime powered iterates under $f(x)=x^d+c$ when $c\in\mathbb{Z}$ is nonzero. Namely, if $f^n(α)=y^p$ for some $n\geq4$, some $α,y\in\mathbb{Z}$, and some prime $p|d$, then $α$ and $y^p$ are necessarily preperiodic and periodic points for $f$ respectively. Moreover, we note that $n=4$ is the smallest possible iterate for which one may make this conclusion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10310 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime-powered images and irreducible polynomials in dynamical semigroups Bhardwaj, Aristaa Boyer-Paulet, Adrian Hindes, Wade Qiu, Emma Sun, Alexander Number Theory Dynamical Systems Let $G=\langle x^d+c_1,\dots,x^d+c_s\rangle$ be a semigroup generated under composition for some $c_1,\dots,c_s\in\mathbb{Z}$ and some $d\geq2$. Then we prove that, outside of an exceptional one-parameter family, $G$ contains a large and explicit subset of irreducible polynomials if and only if it contains at least one irreducible polynomial. In particular, this conclusion holds when $G$ is generated by at least $s\geq3$ polynomials when $d$ is odd and at least $s\geq5$ polynomials when $d$ is even. To do this, we prove a classification result for prime powered iterates under $f(x)=x^d+c$ when $c\in\mathbb{Z}$ is nonzero. Namely, if $f^n(α)=y^p$ for some $n\geq4$, some $α,y\in\mathbb{Z}$, and some prime $p|d$, then $α$ and $y^p$ are necessarily preperiodic and periodic points for $f$ respectively. Moreover, we note that $n=4$ is the smallest possible iterate for which one may make this conclusion. |
| title | Prime-powered images and irreducible polynomials in dynamical semigroups |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2510.10310 |