Exponential sums with reducible polynomials

TL;DR

This paper investigates the distribution of roots of reducible polynomials modulo n and derives asymptotic formulas for exponential sums over these roots, extending previous results known for irreducible polynomials.

Contribution

It provides new asymptotic formulas for exponential sums involving roots of reducible polynomials of degrees 2 and 3, and products of linear factors.

Findings

Asymptotic formulas for exponential sums over roots of reducible quadratic and cubic polynomials.

Extension of uniform distribution results to reducible polynomials.

Analysis of roots of reducible polynomials modulo n.

Abstract

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible polynomials of degree $2$ and $3$ and for finite products of linear factors. In particular, we establish asymptotic formulas for exponential sums over these normalized roots.