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{
    "pk": 64973,
    "title": "Ehrhart quasi-polynomials and parallel translations",
    "subtitle": null,
    "abstract": "Given a rational polytope \\(P \\subset \\mathbb R^d\\), the numerical function counting lattice points in the integral dilations of \\(P\\) is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial \\(\\operatorname{ehr}_P\\) of \\(P\\). In this paper we study the following problem: Given a rational \\(d\\)-polytope \\(P \\subset \\mathbb R^d\\), is there a nice way to know Ehrhart quasi-polynomials of translated polytopes \\(P+  v\\) for all \\( v \\in \\mathbb Q^d\\)? We provide a way to compute such Ehrhart quasi-polynomials using a certain toric arrangement and lattice point counting functions of translated cones of \\(P\\). This method allows us to visualize how constituent polynomials of \\(\\operatorname{ehr}_{P+ v}\\) change in the torus \\(\\mathbb R^d/\\mathbb Z^d\\). We also prove that information of \\(\\operatorname{ehr}_{P+ v}\\) for all \\( v \\in \\mathbb Q^d\\) determines the rational \\(d\\)-polytope \\(P \\subset \\mathbb R^d\\) up to translations by integer vectors, and characterize all rational \\(d\\)-polytopes \\(P \\subset \\mathbb R^d\\) such that \\(\\operatorname{ehr}_{P+ v}\\) is symmetric for all \\( v \\in \\mathbb Q^d\\).Mathematics Subject Classifications: 52C07, 52C35Keywords: Ehrhart quasi-polynomials, rational polytopes, toric arrangements, conic divisorial ideals",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Ehrhart quasi-polynomials"
        },
        {
            "word": "rational polytopes"
        },
        {
            "word": "toric arrangements"
        },
        {
            "word": "conic divisorial ideals"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/0t05r1z0",
    "frozenauthors": [
        {
            "first_name": "Akihiro",
            "middle_name": "",
            "last_name": "Higashitani",
            "name_suffix": "",
            "institution": "Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Osaka, Japan",
            "department": ""
        },
        {
            "first_name": "Satoshi",
            "middle_name": "",
            "last_name": "Murai",
            "name_suffix": "",
            "institution": "Department of Mathematics, Faculty of Education, Waseda University, Tokyo, Japan",
            "department": ""
        },
        {
            "first_name": "Masahiko",
            "middle_name": "",
            "last_name": "Yoshinaga",
            "name_suffix": "",
            "institution": "Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Japan",
            "department": ""
        }
    ],
    "date_submitted": "2025-03-14T16:55:36Z",
    "date_accepted": "2025-03-14T16:55:36Z",
    "date_published": "2025-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64973/galley/49783/download/"
        }
    ]
}