{
  "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 $f(X)=X^n+A(BX+1)^m,$ where $1\\le m<n$, $\\gcd(n,mB)=1$, and $A,B\\in\\mathbb{Z}\\setminus\\{0\\}$. We show that $f$ is monogenic if and only if both $A~\\text{and}~n^n+(-1)^{n+m}B^n(n-m)^{\\,n-m}m^mA$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36811",
  "author_total": 2,
  "authors": [
    {
      "name": "Michail Karatarakis"
    },
    {
      "name": "Sumandeep Kaur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1qv5kjv5bbsg7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36811",
  "fulltext_url": null,
  "id": "1qv5kjv5bbsg7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36811",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214928663",
  "pdf_url": "https://arxiv.org/pdf/2609.36811",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1qv5kjv5bbsg7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36811",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36811"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Local Approach to Monogenity with an Application to Lenny Jones' Conjecture",
  "title_text": "A Local Approach to Monogenity with an Application to Lenny Jones' Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/1qv5kjv5bbsg7",
  "venue": "arXiv",
  "versions": []
}