{
  "id": "1410.1600",
  "version": "v1",
  "published": "2014-10-07T02:29:30.000Z",
  "updated": "2014-10-07T02:29:30.000Z",
  "title": "There are no two non-real conjugates of a Pisot number with the same imaginary part",
  "authors": [
    "Artūras Dubickas",
    "Kevin G. Hare",
    "Jonas Jankauskas"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that the number $\\alpha=(1+\\sqrt{3+2\\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4$ satisfy the additive relation $\\alpha_1+\\alpha_2=\\alpha_3+\\alpha_4$. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations $\\alpha_1 = \\alpha_2 + \\alpha_3+\\alpha_4$ or $\\alpha_1 + \\alpha_2 + \\alpha_3 + \\alpha_4 =0$ cannot be solved in conjugates of a Pisot number $\\alpha$. We also show that the roots of the Siegel's polynomial $x^3-x-1$ are the only solutions to the three term equation $\\alpha_1+\\alpha_2+\\alpha_3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $\\alpha_1=\\alpha_2+\\alpha_3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-07T02:29:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pisot number",
      "imaginary part",
      "non-real conjugates",
      "term equation",
      "minimal polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1600D"
    }
  }
}