PreprintCC BY 4.0arXiv
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 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.
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BibTeX
@misc{ceballos2026squarefree,
title = {{On the Squarefree Values of Degree-$2q$ Polynomials}},
author = {Sergio Ricardo Zapata Ceballos and Fatemeh Jalalvand},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.10335},
eprint = {2608.10335},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.10335},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Squarefree Values of Degree-2q Polynomials
AU - Sergio Ricardo Zapata Ceballos
AU - Fatemeh Jalalvand
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.10335
UR - https://arxiv.org/abs/2608.10335
AB - 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 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.
ER -