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

In this first article of a multi-part series, 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 elegant counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics, and also by interesting point-counting result of Narkiewicz on rational periodic points of any odd degree map $\varphi_{d, c}$ in arithmetic dynamics, we then first prove that for any prime $p\geq 3$, the average number of distinct integral fixed points of any $\varphi_{p, c}$ modulo $p$ is $3$ or $0$ as $c$ tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of $\varphi_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any $\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings.