{
  "annotation_id": "21cb821a-515e-472d-8ee8-124f9b104d5b",
  "date_created": "2026-02-17T05:53:05.054000Z",
  "date_modified": "2026-02-17T05:53:05.054000Z",
  "file_hash": "1ed6f3a01509dfe72c0c483668386825c1c6fcd2663f0245874c058cfb9996db",
  "private": false,
  "record": {
    "abstract": "Suppose that $f(x) \\in \\mathbb{R}[x_1,\\dots, x_n]$ and $g(x) \\in \\mathbb{R}[x_1,\\dots, x_n]$ are two real polynomials of degree $d$ in $n$ variables. If the polynomials $f$ and $g$ are the same up to orthogonal symmetry a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element $R\\in O(n)$ such that $f(Rx)=g(x)$. One may directly solve this problem by constructing a nonlinear system of equations induced by the relation $f(Rx)=g(x)$ along with the identities of the orthogonal group however this approach becomes quite computationally expensive for larger values of $n$ and $d$. To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analysis (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the $R\\in O(n)$ such that $f(Rx)=g(x)$.",
    "arxiv_id": "2601.06148",
    "authors": [
      "Martin Helmer",
      "David Hong",
      "Hoon Hong"
    ],
    "categories": [
      "math.RA",
      "cs.NA",
      "math.AC",
      "math.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis",
    "url": "https://arxiv.org/abs/2601.06148",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "dea7b956-ad76-4cc4-9f4e-2027a7c554c2",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}