Statistical distribution of roots of a polynomial modulo primes

Abstract

Let $f(x)=x^n+a_{n-1}{x}^{n-1}+\cdots +a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\mathrm{mod}p$ with $0\le r_1\le \cdots \le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil {\sum }_{i\in S}{r}_i/p\rceil$ for a subset $S$ of $\{1,\dots ,n\}$ when $p\to \infty$.