{
  "id": "2110.14235",
  "version": "v2",
  "published": "2021-10-27T07:39:57.000Z",
  "updated": "2022-12-20T06:49:46.000Z",
  "title": "Algebraic intersection in regular polygons",
  "authors": [
    "Julien Boulanger",
    "Erwan Lanneau",
    "Daniel Massart"
  ],
  "comment": "New version with the first author added and completely different methods. We focus on the $n=2m+1$ case, the $n=4m$ case is dealt with in a forthcoming paper by the first author. 30pages, 15 figures",
  "categories": [
    "math.DS",
    "math.DG"
  ],
  "abstract": "We study the function $$\\mbox{KVol} : (X,\\omega)\\mapsto \\mbox{Vol} (X,\\omega) \\sup_{\\alpha,\\beta} \\frac{\\mbox{Int} (\\alpha,\\beta)}{l_g (\\alpha) l_g (\\beta)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\\mathcal T_n$ be the Teichm\\\"uller discs of the original Veech surface $(X_n,\\omega_n)$ arising from right-angled triangle with angles $(\\pi/2,\\pi/n,(n-2)\\pi/2n)$ by the unfolding construction for $n\\geq 5$. For $n \\equiv 1 \\mod 2$ and any $(X,\\omega)\\in \\mathcal T_n$, we establish the (sharp) bounds $$ \\frac{n}{2} \\cot \\frac{\\pi}{n} \\leq \\mbox{KVol}(X,\\omega) \\leq \\frac{n}{2} \\cot \\frac{\\pi}{n} \\cdot \\frac1{\\sin \\frac{2\\pi}{n}}.$$ The lower bound is uniquely realized at $(X_n,\\omega_n)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-12-20T06:49:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D40",
      "32G15",
      "53C22"
    ],
    "keywords": [
      "regular polygons",
      "algebraic intersection",
      "original veech surface",
      "moduli spaces",
      "translation surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}