On Mahler's inequality and small integral generators of totally complex number fields

TL;DR

This paper refines Mahler's inequality for a specific class of complex polynomials, providing sharper bounds on the Mahler measure related to the discriminant and degree, with implications for generators of totally complex number fields.

Contribution

It improves Mahler's lower bound exponent for a special polynomial class and demonstrates the bound's sharpness, extending results to totally complex number fields.

Findings

Enhanced Mahler's lower bound with exponent 1/(2d-3)

Sharpness of the new bound established

Improved bounds for integral generators of totally complex number fields

Abstract

We improve Mahler's lower bound for the Mahler measure in terms of the discriminant and degree for a specific class of polynomials: complex monic polynomials of degree $d\geq 2$ such that all roots with modulus greater than some fixed value $r\geq1$ occur in equal modulus pairs. We improve Mahler's exponent $\frac{1}{2d-2}$ on the discriminant to $\frac{1}{2d-3}$. Moreover, we show that this value is sharp, even when restricting to minimal polynomials of integral generators of a fixed not totally real number field. An immediate consequence of this new lower bound is an improved lower bound for integral generators of number fields, generalising a simple observation of Ruppert from imaginary quadratic to totally complex number fields of arbitrary degree.