On the Squarefree Values of Degree-$2q$ Polynomials

Abstract

In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.

arXiv commentThe proof of Proposition 2.3 is incomplete. The argument does not provide the estimate needed to control the contribution of square divisors arising from prime ideals of large norm. Consequently, the claimed squarefreeness result is not established. The authors currently do not know how to complete the proof