{
  "abstract": "Let $P$ be a $d$-dimensional lattice polytope with Ehrhart polynomial $L_P(t)$. Motivated by the study of Ehrhart positivity and magic positivity, we investigate the Taylor coefficients $\\mathsf{A}_j(P;k)$ in the shifted expansion $L_P(t)=\\sum_{j=0}^{d}\\mathsf{A}_j(P;k)(t-k)^j$ about a real center $k$. In this paper, we obtain the following four main results. (i) We give exact formulas for these coefficients in terms of the ordinary Ehrhart coefficients, the $h^*$-vector, elementary symmetric functions, and Stirling numbers. (ii) We denote by $τ(P)$ and $τ^+(P)$ the smallest nonnegative integral centers at which all Taylor coefficients are nonnegative and positive, respectively. If $s$ is the degree of the $h^*$-polynomial, then $0\\leqτ(P)\\leqτ^+(P)\\leq\\min\\{\\max\\{0,s-1\\},\\lfloor\\frac{d-1}{2}\\rfloor\\}$. As an application, we slightly improve an upper bound due to Beck, De Loera, Develin, Pfeifle, and Stanley. That is, every real root of $L_P(t)$ lies in $[-d,\\lfloor\\frac{d-1}{2}\\rfloor)$. (iii) Let $ρ(P)$ be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If $λ_{\\mathbb{R}}(f)$ denotes the largest real zero of $f(t)$, with value $-\\infty$ when no such zero exists, then $ρ(P)=\\max\\{0,\\max_{0\\leq j<d}λ_{\\mathbb{R}}\\!(L_P^{(j)})\\}$. (iv) We establish structural properties of the Taylor coefficients $\\mathsf{A}_j(P;k)$, including derivative interlacing, palindromic reflection symmetries, and Laguerre and Newton inequalities. As a final note, these results provide a systematic partial answer to an open problem listed on the website of the American Institute of Mathematics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03327",
  "author_total": 2,
  "authors": [
    {
      "name": "Feihu Liu"
    },
    {
      "name": "Zihao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/56tvh5d30ef8j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03327",
  "fulltext_url": null,
  "id": "56tvh5d30ef8j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03327",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7208753407",
  "pdf_url": "https://arxiv.org/pdf/2609.03327",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/56tvh5d30ef8j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.03327",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03327"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Taylor Positivity of Ehrhart Polynomials",
  "title_text": "Taylor Positivity of Ehrhart Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/56tvh5d30ef8j",
  "venue": "arXiv",
  "versions": []
}