On the number of integer polynomials with multiplicatively dependent roots

TL;DR

This paper investigates the quantity of integer polynomials with fixed degree and bounded height that have roots which are multiplicatively dependent, providing bounds and asymptotic formulas for these counts.

Contribution

It offers new bounds and asymptotic formulas for counting integer polynomials with multiplicatively dependent roots, advancing understanding in this area.

Findings

Sharp lower bounds established

Upper bounds derived

Asymptotic formulas provided

Abstract

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.