{
  "id": "2112.05004",
  "version": "v2",
  "published": "2021-12-09T15:58:04.000Z",
  "updated": "2022-05-16T05:20:39.000Z",
  "title": "Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers",
  "authors": [
    "Cheng-Chao Huang"
  ],
  "categories": [
    "math.NT",
    "cs.CC",
    "cs.SC"
  ],
  "abstract": "In this paper, we study linear forms \\[\\lambda = \\beta_1\\mathrm{e}^{\\alpha_1}+\\cdots+\\beta_m\\mathrm{e}^{\\alpha_m},\\] where $\\alpha_i$ and $\\beta_i$ are algebraic numbers. An explicit lower bound for the absolute value of $\\lambda$ is proved, which is derived from \"th\\'eor\\`eme de Lindemann--Weierstrass effectif\" via constructive methods in algebraic computation. Besides, the existence of $\\lambda$ with an explicit upper bound is established on the result of counting algebraic numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-05-16T05:20:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "explicit bounds",
      "exponentials",
      "explicit upper bound",
      "study linear forms",
      "explicit lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}