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{
    "pk": 65012,
    "title": "A degree bound for planar functions",
    "subtitle": null,
    "abstract": "Using Stickelberger's theorem on Gauss sums, we show that if \\(F\\) is a planar function on a finite field \\(\\mathbb{F}_q\\), then for all non-zero functions \\(G : \\mathbb{F}_q \\to \\mathbb{F}_q\\), we have \\begin{equation*} d_{\\mathsf{alg}}(G \\circ F) - d_{\\mathsf{alg}}(G) \\le \\frac{n(p-1)}{2}, \\end{equation*} where \\(q = p^n\\) with \\(p\\) a prime and \\(n\\) a positive integer, and \\(d_{\\mathsf{alg}}(F)\\) is the algebraic degree of \\(F\\), i.e., the maximum degree of the corresponding system of \\(n\\) lowest-degree interpolating polynomials for \\(F\\) considered as a function on \\(\\mathbb{F}_p^n\\). This bound implies the (known) classification of planar polynomials over \\(\\mathbb{F}_p\\) and planar monomials over \\(\\mathbb{F}_{p^2}\\). As a new result, using the same degree bound, we complete the classification of planar monomials for all \\(n = \\smash{2^k}\\) with \\(p›5\\) and \\(k\\) a non-negative integer. Finally, we state a conjecture on the sum of the base-\\(p\\) digits of integers modulo \\(q-1\\) that implies the complete classification of planar monomials over finite fields of characteristic \\(p›5\\).\n \nMathematics Subject Classifications: 05B25, 11T06, 11T24\n \nKeywords: Planar function, algebraic degree, Stickelberger's theorem, digit sum",
    "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": "Planar function"
        },
        {
            "word": "algebraic degree"
        },
        {
            "word": "Stickelberger's theorem"
        },
        {
            "word": "digit sum"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/6697p4nj",
    "frozenauthors": [
        {
            "first_name": "Christof",
            "middle_name": "",
            "last_name": "Beierle",
            "name_suffix": "",
            "institution": "Faculty of Computer Science, Ruhr University Bochum, Bochum, Germany",
            "department": ""
        },
        {
            "first_name": "Tim",
            "middle_name": "",
            "last_name": "Beyne",
            "name_suffix": "",
            "institution": "Faculty of Computer Science, Ruhr University Bochum, Bochum, Germany -- COSIC, KU Leuven, Leuven, Belgium",
            "department": ""
        }
    ],
    "date_submitted": "2025-09-12T13:25:43Z",
    "date_accepted": "2025-09-12T13:25:43Z",
    "date_published": "2025-09-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65012/galley/49822/download/"
        }
    ]
}