A Local Approach to Monogenity with an Application to Lenny Jones' Conjecture
Abstract
The study of monogenic polynomials is a classical problem in algebraic number theory. Existing criteria for deciding whether a polynomial is monogenic typically rely on discriminant computations together with methods such as Dedekind's criterion, Newton polygons, or valuation-theoretic techniques. In this paper, we develop a general local criterion for the p-maximality of orders generated by roots of arbitrary monic irreducible polynomials. As an application, we apply this to irreducible polynomials of the type f(X)=Xn+A(BX+1)m, where 1≤ m<n, gcd(n,mB)=1, and A,B∈ℤ∖{0}. We show that f is monogenic if and only if both A and nn+(−1)n+mBn(n−m) n−mmmA are square-free. This provides a new proof of the main theorem of \cite{KK}, thereby proving Lenny Jones' conjecture \cite[Conjecture 4.1]{LJ}. Furthermore, we obtain explicit infinite families of irreducible non-monogenic polynomials, including trinomial, quadrinomial, and power-compositional families.
Related
- Combination Theorems and the Farrell--Jones Conjecture for Graph Products of Pairs
- Monogenity of the irreducible factors of (k,t)-Fibonacci and (k,t)-Lucas polynomials
- Higher-order jet and its application to the Novikov conjecture
- Idempotence in Carmichael and Giuga numbers and its application to the Agoh-Giuga conjecture
- Asymptotics of local height pairing without Hodge conjecture
BibTeX
@misc{karatarakis2026local,
title = {{A Local Approach to Monogenity with an Application to Lenny Jones' Conjecture}},
author = {Michail Karatarakis and Sumandeep Kaur},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36811},
eprint = {2609.36811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Local Approach to Monogenity with an Application to Lenny Jones' Conjecture
AU - Michail Karatarakis
AU - Sumandeep Kaur
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36811
UR - https://arxiv.org/abs/2609.36811
AB - The study of monogenic polynomials is a classical problem in algebraic number theory. Existing criteria for deciding whether a polynomial is monogenic typically rely on discriminant computations together with methods such as Dedekind's criterion, Newton polygons, or valuation-theoretic techniques. In this paper, we develop a general local criterion for the p-maximality of orders generated by roots of arbitrary monic irreducible polynomials. As an application, we apply this to irreducible polynomials of the type f(X)=X^n+A(BX+1)^m, where 1≤ m<n, gcd(n,mB)=1, and A,B∈ℤ∖{0}. We show that f is monogenic if and only if both A and n^n+(−1)^n+mB^n(n−m)^ n−mm^mA are square-free. This provides a new proof of the main theorem of \cite{KK}, thereby proving Lenny Jones' conjecture \cite[Conjecture 4.1]{LJ}. Furthermore, we obtain explicit infinite families of irreducible non-monogenic polynomials, including trinomial, quadrinomial, and power-compositional families.
ER -