A statistical relation of roots of a polynomial in different local fields. III (Q436051)

This paper is the continuation of previous work of the author [in: Number theory. Dreaming in dreams. Proceedings of the 5th China-Japan seminar, Higashi-Osaka, Japan, 2008. Hackensack, NJ: World Scientific. Series on Number Theory and Its Applications 6, 106--126 (2010; Zbl 1217.11029); Math. Comput. 78, No. 265, 523--536 (2009; Zbl 1214.11127)]. Let NEWLINE\[NEWLINEf(x)=x^ n+a_{n-1}x^{n-1}+\cdots+ a_ 0\in {\mathbb Z}[x]NEWLINE\]NEWLINE be a monic polynomial and let NEWLINE\[NEWLINE \mathrm{Spl}(f):=\{p\mid f(x)\bmod p \text{\;decomposes fully}\}, NEWLINE\]NEWLINE where \(p\) denotes a rational prime. For \(p\in \mathrm{Spl}(f)\), let \(r_ 1,\ldots, r_ n\in {\mathbb Z}\), \(0\leq r_ i\leq p-1\) be the solutions of \(f(x)\equiv 0\bmod p\) for \(p\in \mathrm{Spl}(f)\). We have that NEWLINE\[NEWLINEa_ {n-1} +\sum _{i=1}^ n r_ i\equiv 0\bmod p.NEWLINE\]NEWLINE Let \(C_ p(f) \in {\mathbb Z}\) be such that NEWLINE\[NEWLINEa_ {n-1}+\sum_{i=1}^ n r_ i = C_ p(f) p.NEWLINE\]NEWLINE We have the natural density NEWLINE\[NEWLINE \mathrm{Pr}(k,f,x):={{|\{p\in\mathrm{Spl}(f)\mid p\leq x, C_ p(f)=k\}|}\over{|\{p\in\mathrm{Spl}(f)\mid p\leq x\}|}}\quad \text{and}\quad \mathrm{Pr}(k,f):=\lim_{x\to \infty}\mathrm{Pr}(k,f,x). NEWLINE\]NEWLINE In this paper the author studies a statistical relation of the roots of \(f(x) \bmod p\) in several cases where \(f\) is an irreducible polynomial and \(f\) has two different decompositions \(f(x)=g(h(x))\) such that \(g,h\) are monic polynomials over \({\mathbb Z}\) with \(h(0)=0\), \(1<\deg h < \deg f\). Also, a series of polynomials \(f\) with two nontrivial decompositions \(f(x)=g(h(x))\) are constructed.