# Root separation for reducible monic polynomials of odd degree

## TL;DR

This paper investigates the minimal root separation of reducible monic integer polynomials of odd degree, establishing upper bounds and improving lower bounds for specific degrees, advancing understanding of root distribution.

## Contribution

It introduces new bounds for the exponent governing root separation in reducible monic polynomials of odd degree, refining previous estimates and expanding theoretical knowledge.

## Key findings

- e_r*(d) <= d-2 for all odd degrees d
- Improved lower bounds for e_r*(d) when d=5,7,9
- Enhanced understanding of root separation behavior in reducible polynomials

## Abstract

We study root separation of reducible monic integer polynomials of odd degree. Let h(P) be the naive height, sep(P) the minimal distance between two distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let e_r*(d)=limsup_{deg(P)=d, h(P)-> +infty} e(P), where the limsup is taken over the reducible monic integer polynomials P(x) of degree d. We prove that e_r*(d) <= d-2. We also obtain a lower bound for e_r*(d) for d odd, which improves previously known lower bounds for e_r*(d) when d = 5, 7, 9.