{
  "abstract": "Let $\\mathbb F_q$ be a finite field and consider quadratic polynomials $f(X)=X^2+bX+c\\in\\mathbb F_q[X]$ with primitive constant term $c$. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive $c\\in\\mathbb F_q$, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients $b$ for which $X^2+bX+c$ is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the $(q+1)$-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable $B$, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over $A=\\mathbb F_q[C]/(Φ_{q-1}(C))$. We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of $A[B]$. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.01327",
  "author_total": 2,
  "authors": [
    {
      "name": "Li Zhu"
    },
    {
      "name": "Hongfeng Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6qhrccmrp8bqn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.01327",
  "fulltext_url": null,
  "id": "6qhrccmrp8bqn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.01327",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7172436578",
  "pdf_url": "https://arxiv.org/pdf/2608.01327",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6qhrccmrp8bqn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.01327",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.01327"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields",
  "title_text": "The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/6qhrccmrp8bqn",
  "venue": "arXiv",
  "versions": []
}