Prime-powered images and irreducible polynomials in dynamical semigroups
Aristaa Bhardwaj, Adrian Boyer-Paulet, Wade Hindes, Emma Qiu, Alexander Sun
TL;DR
The paper investigates when semigroups generated by unicritical polynomials $\{x^d+c_i\}$ contain a positive proportion of irreducible polynomials over $\mathbb{Q}$. It develops a dynamical $p$th-power classification for maps $f(x)=x^d+c$ with $c\in\mathbb{Z}$, proving that if a sufficiently high iterate yields a $p$th power, then the initial point is preperiodic and the $p$th-power value is periodic; this classification is sharp with minimal iterate $N$ depending on $d$. Leveraging this, the authors show stability of irreducibility under iteration and, outside a small exceptional one-parameter family, that $G$ contains a positive proportion of irreducibles iff it contains at least one irreducible polynomial, with explicit irreducible constructions tailored to the parity of $d$ and the special forms of the coefficients. The results provide a structural link between dynamical properties of $f(x)=x^d+c$ and arithmetic properties of polynomial irreducibility in semigroups, advancing understanding of density of irreducibles in dynamical settings.
Abstract
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.