A statistical relation of roots of a polynomial in different local fields III

Abstract

Let $f(x)$ be a monic polynomial in $\mathbb{Z}[x]$. We have observed a statistical relation of roots of $f(x) \bmod p$ for different primes $p$, where $f(x)$ decomposes completely modulo $p$. We could guess what happens if $f(x)$ is irreducible and has at most one decomposition $f(x) = g(h(x))$ such that $g,h$ are monic polynomials over $\mathbb{Z}$ with $h(0) = 0$, $1 < \deg h < \deg f$. In this paper, we study cases that $f$ has two different such decompositions. Besides, we construct a series of polynomials f which have two non-trivial different decompositions $f(x) = g(h(x))$.