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{
    "pk": 64995,
    "title": "Mixed radix numeration bases: Horner's rule, Yang-Baxter equation and Furstenberg's conjecture",
    "subtitle": null,
    "abstract": "Mixed radix bases in numeration is a very old notion but it is rarely studied on its own or in relation with concrete problems related to number theory. Starting from the natural question of the conversion of a basis to another for integers as well as polynomials, we use mixed radix bases to introduce two-dimensional arrays with suitable filling rules. These arrays provide algorithms of conversion which use only a finite number of Euclidean division to convert from one basis to another; it is interesting to note that these algorithms are generalizations of the well-known Horner's rule of quick evaluation of polynomials. The two-dimensional arrays with local transformations are reminiscent of statistical mechanics models: we show that changes between three numeration bases are related to the set-theoretical Yang-Baxter equation and this is, up to our knowledge, the first time that such a structure is described in number theory. As an illustration, we reinterpret well-known results around Furstenberg's conjecture in terms of Yang-Baxter transformations between mixed radix bases, hence opening the way to alternative approaches.\n \nMathematics Subject Classifications: 11A63, 11A67, 16T25\n \nKeywords: Numeration basis, Yang-Baxter equation",
    "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": "Numeration basis"
        },
        {
            "word": "Yang-Baxter equation"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/8jj0b04v",
    "frozenauthors": [
        {
            "first_name": "Damien",
            "middle_name": "",
            "last_name": "Simon",
            "name_suffix": "",
            "institution": "Sorbonne Université and Université Paris Cité, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, 4, place Jussieu, F-75005 Paris, France",
            "department": ""
        }
    ],
    "date_submitted": "2025-07-16T00:40:19+09:00",
    "date_accepted": "2025-07-16T00:40:19+09:00",
    "date_published": "2025-07-15T16:00:00+09:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64995/galley/49805/download/"
        }
    ]
}