{
  "id": "2412.19335",
  "version": "v1",
  "published": "2024-12-26T19:13:17.000Z",
  "updated": "2024-12-26T19:13:17.000Z",
  "title": "Moduli spaces of polynomial maps and multipliers at small cycles",
  "authors": [
    "Valentin Huguin"
  ],
  "comment": "63 pages",
  "categories": [
    "math.DS",
    "math.AG"
  ],
  "abstract": "Fix an integer $d \\geq 2$. The space $\\mathcal{P}_{d}$ of polynomial maps of degree $d$ modulo conjugation by affine transformations is naturally an affine variety over $\\mathbb{Q}$ of dimension $d -1$. For each integer $P \\geq 1$, the elementary symmetric functions of the multipliers at all the cycles with period $p \\in \\lbrace 1, \\dotsc, P \\rbrace$ induce a natural morphism $\\operatorname{Mult}_{d}^{(P)}$ defined on $\\mathcal{P}_{d}$. In this article, we show that the morphism $\\operatorname{Mult}_{d}^{(2)}$ induced by the multipliers at the cycles with periods $1$ and $2$ is both finite and birational onto its image. In the case of polynomial maps, this strengthens results by McMullen and by Ji and Xie stating that $\\operatorname{Mult}_{d}^{(P)}$ is quasifinite and birational onto its image for all sufficiently large integers $P$. Our result arises as the combination of the following two statements: $\\mathord{\\bullet}$ A sequence of polynomials over $\\mathbb{C}$ of degree $d$ with bounded multipliers at its cycles with periods $1$ and $2$ is necessarily bounded in $\\mathcal{P}_{d}(\\mathbb{C})$. $\\mathord{\\bullet}$ A generic conjugacy class of polynomials over $\\mathbb{C}$ of degree $d$ is uniquely determined by its multipliers at its cycles with periods $1$ and $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-26T19:13:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F46",
      "37P45",
      "37F10",
      "37P05",
      "37P30"
    ],
    "keywords": [
      "polynomial maps",
      "small cycles",
      "multipliers",
      "moduli spaces",
      "generic conjugacy class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}