{
  "annotation_id": "dae138b7-1c63-46e5-942f-315b13e5c430",
  "date_created": "2026-02-17T05:53:08.863000Z",
  "date_modified": "2026-02-17T05:53:08.863000Z",
  "file_hash": "e354a328a28856497737d03412e70b833be614af63e0e01d8e0646c7ebbfe1b0",
  "private": false,
  "record": {
    "abstract": "Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We conduct a systematic study of UP-equivalence, which leads to the first average-case quasi-polynomial time algorithm, called \\texttt{HEM}, for deciding the unimodular equivalence of $d$-dimensional integral simplices, as well as achieving a polynomial-time complexity with a failure probability less than $2.5 \\times 10^{-7}$. A key ingredient is the introduction of the \\emph{permuted Hermite normal form} and its associated \\emph{pattern group}, which streamlines the UP-equivalence test by comparing canonical forms derived from induced coset representatives. We also present an acceleration strategy based on Smith normal forms. As a theoretical by-product, we prove that two full-dimensional integral simplices are unimodularly equivalent if and only if their $n$-dimensional pyramids are unimodularly equivalent. This resolves an open question posed by Abney-McPeek et al.",
    "arxiv_id": "2601.06819",
    "authors": [
      "Feihu Liu",
      "Sihao Tao",
      "Guoce Xin"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Unimodular Equivalence of Integral Simplices",
    "url": "https://arxiv.org/abs/2601.06819",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "2b0e6145-dd2b-4784-b0de-4847422f099c",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}