Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I

Abstract: In this article, we inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}$ and the coefficient $c$, where $d > 2$ is an integer. Inspired greatly by the elegance of the counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics and by an interesting point-counting result of Narkiewicz on rational periodic points of $\varphi_{d, c}$ for any odd degree $d>2$ in arithmetic dynamics, we prove that for any given prime integer $p\geq 3$, the average number of distinct integral fixed points of any $\varphi_{p, c}$ modulo $p$ is 3 or 0, as $p$ tends to infinity. Moreover, we also show that a density of $0\%$ of integer polynomials $\varphi_{p, c}(x)$ have three fixed points modulo $p$, as $c$ tends to infinity. Consequently, a density of $100\%$ of integer polynomials $f(x): = x^p -x + c$ induces odd prime degree-$p$ number fields $K_{f}:=\mathbb{Q}[x]\slash (f(x))$. Since every number field $K_{f}$ comes naturally with the ring $\mathcal{O}{K{f}}$ of integers, applying a density result of Bhargava-Shankar-Wang on our irreducible integer polynomials $f(x)$ yields that a density equal to $\zeta(2)^{-1}$ of monic polynomials $f(x)$ is such that $\mathbb{Z}[x]\slash (f(x))$ is the ring $\mathcal{O}{K{f}}$ of integers. Motivated by a conjecture of Hutz on rational periodic points of $\varphi_{p-1, c}$ for any given prime $p\geq 5$ in arithmetic dynamics, we then also prove unconditionally that the average number of distinct integral fixed points of any $\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$, as $p$ tends to infinity. We again use the same counting machinery as in odd prime degree setting to then also deduce several statistical results.