{
  "id": "2202.11740",
  "version": "v1",
  "published": "2022-02-23T19:03:36.000Z",
  "updated": "2022-02-23T19:03:36.000Z",
  "title": "Lower bounds on the rank and symmetric rank of real tensors",
  "authors": [
    "Kexin Wang",
    "Anna Seigal"
  ],
  "comment": "26 pages, 3 figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, there was only one real counterexample known, constructed in a paper of Shitov. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space involving larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-23T19:03:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A69",
      "14N07",
      "13P15"
    ],
    "keywords": [
      "lower bound",
      "real tensor",
      "real symmetric rank differ",
      "real rank",
      "comons conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}