{
  "id": "2502.03154",
  "version": "v1",
  "published": "2025-02-05T13:24:21.000Z",
  "updated": "2025-02-05T13:24:21.000Z",
  "title": "Infinite products with algebraic numbers",
  "authors": [
    "Simon Kristensen",
    "Mathias Løkkegaard Laursen"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We obtain general criteria for giving a lower bound on the degree of numbers of the form $\\prod_{n=1}^\\infty \\left(1+\\frac{b_n}{\\alpha_n}\\right)$ or of the form $\\prod_{m=1}^\\infty \\left(1+ \\sum_{n=1}^\\infty \\frac{b_{n,m}}{\\alpha_{n,m}}\\right)$, where the $\\alpha_n$ and $\\alpha_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-05T13:24:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72",
      "11J81"
    ],
    "keywords": [
      "algebraic numbers",
      "infinite products",
      "lower bound",
      "general criteria",
      "growth conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}