{
  "abstract": "Classical Ehrhart theory measures the discrete capacity of a convex rational (or integral) polytope $P$ by counting the number of lattice points in the $t$-th dilate $tP$ of $P$. In this paper, we extend this paradigm by replacing a lattice point with a geometric object $Q$ of dimension at most $\\dim P$. We show that the counting function $\\mathrm{ehr}(P,Q;t)$ of such valid translations of $Q$ into $tP$ inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term $\\mathrm{vol}(P)t^d$, where $d = \\dim P$. This result is naturally derived by induction on the dimension, based on the classical quasi-polynomiality (or polynomiality) of Ehrhart functions. Similarly, replacing $P$ with its relative interior $P^\\circ$ defines $\\mathrm{ehr}^\\circ(P,Q;t)$, which is also shown to be an eventual quasi-polynomial (or eventual polynomial). Furthermore, we prove several formulas for relative Ehrhart functions under the condition that $P = kP_0$ and $Q = lQ_0$ for some integers $k$, $l > 0$, and polytopes $P_0$ and $Q_0$ with $\\dim Q_0 > 0$ such that $Q_0$ is inscribed in $P_0$. In particular, we prove that under this condition, $\\mathrm{ehr}(P,Q;-t) = (-1)^{d}\\mathrm{ehr}^\\circ (P,Q;t+ρ)$ $(t \\gg 0)$ holds for some integer $ρ> 0$ if and only if $k \\mid 2l$ via the classical Ehrhart--Macdonald reciprocity law. We also prove that under the same condition, $\\mathrm{ehr}(P,Q;t) = \\mathrm{ehr}^\\circ (P,Q;t+σ)$ $(t \\gg 0)$ holds for some integer $σ> 0$ if and only if $\\mathrm{ehr}^\\circ (P_0;\\mathrm{codeg}\\,P_0) = 1$ and $k \\mid \\mathrm{codeg}\\,P_0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.10370",
  "author_total": 1,
  "authors": [
    {
      "name": "Takashi Hirotsu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9xk71zmzgqfr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.10370",
  "fulltext_url": null,
  "id": "9xk71zmzgqfr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.10370",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7202245350",
  "pdf_url": "https://arxiv.org/pdf/2608.10370",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9xk71zmzgqfr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2608.10370",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.10370"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Eventual polynomiality and dualities of relative Ehrhart functions",
  "title_text": "Eventual polynomiality and dualities of relative Ehrhart functions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/9xk71zmzgqfr9",
  "venue": "arXiv",
  "versions": []
}