2023/08/09 by Murray Child, Child, Murray, Martin Widmer +1 · 1 citation
Computer Science · Mathematics · #11G50 #11R04 #11R06 #30C10 #Analytic Number Theory Research #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2308.05188
openalex publication_date 2023/08/09 · openalex created_date 2023/08/12 · openalex updated_date 2026/07/28
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≥ 2 such that all roots with modulus greater than some fixed value r≥1 occur in equal modulus pairs. We improve Mahler's exponent (1)/(2d-2) on the discriminant to (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.