The proportion of $k$-cycles for polynomials modulo primes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910627537092608 |
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| author | Root, Jonathan |
| author_facet | Root, Jonathan |
| contents | Let $f(x) \in \mathbb{F}_p[x]$, and define the orbit of $x\in \mathbb{F}_p$ under the iteration of $f$ to be the set \[ \mathcal{O}(x):=\{x,f(x),(f\circ f)(x),(f\circ f\circ f)(x),\dots\}. \] An orbit is a $k$-cycle if it is periodic of length $k$. In this paper we fix a polynomial $f(x)$ with integer coefficients and for each prime $p$ we consider $f(x) \pmod p$ obtained by reducing the coefficients of $f(x)$ modulo $p$. We ask for the density of primes $p$ such that $f(x)\pmod p$ has a $k$-cycle in $\mathbb{F}_p$. We prove that in many cases the density is at most $1/k$. We also give an infinite family of polynomials in each degree with this property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The proportion of $k$-cycles for polynomials modulo primes Root, Jonathan Number Theory Dynamical Systems 37P35, 37P15, 37P05, 11R09 Let $f(x) \in \mathbb{F}_p[x]$, and define the orbit of $x\in \mathbb{F}_p$ under the iteration of $f$ to be the set \[ \mathcal{O}(x):=\{x,f(x),(f\circ f)(x),(f\circ f\circ f)(x),\dots\}. \] An orbit is a $k$-cycle if it is periodic of length $k$. In this paper we fix a polynomial $f(x)$ with integer coefficients and for each prime $p$ we consider $f(x) \pmod p$ obtained by reducing the coefficients of $f(x)$ modulo $p$. We ask for the density of primes $p$ such that $f(x)\pmod p$ has a $k$-cycle in $\mathbb{F}_p$. We prove that in many cases the density is at most $1/k$. We also give an infinite family of polynomials in each degree with this property. |
| title | The proportion of $k$-cycles for polynomials modulo primes |
| topic | Number Theory Dynamical Systems 37P35, 37P15, 37P05, 11R09 |
| url | https://arxiv.org/abs/2410.00716 |