{
  "id": "2202.04452",
  "version": "v1",
  "published": "2022-02-09T13:26:42.000Z",
  "updated": "2022-02-09T13:26:42.000Z",
  "title": "On the trace of linear combination of powers of algebraic numbers",
  "authors": [
    "Aprameyo Pal",
    "Veekesh Kumar",
    "R. Thangadurai"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we prove three main results. Let $\\lambda_1, \\ldots, \\lambda_k$ and $\\alpha_1, \\ldots, \\alpha_k$ be nonzero algebraic numbers for some integer $k\\geq 1$ and let $L = \\mathbb{Q}(\\lambda_1, \\ldots, \\lambda_k, \\alpha_1, \\ldots, \\alpha_k)$ be the number field. Let $K$ be the Galois closure of $L$ and let $h$ be the order of the torsion subgroup of $K^\\times$. We first prove an extension of a result of B. de Smit [3] as follows: {\\it Take $\\lambda_i = b_i \\in\\mathbb{Q}$ for $i=1,\\ldots, k$ such that $b_1+\\cdots+b_k = n\\ne 0$ and $\\alpha_j$'s are some of the Galois conjugates (not necessarily distinct) of $\\alpha_1$ for all $j = 2, \\ldots, k$ and $d\\geq 1$ is the degree of $\\alpha_1$. If $\\displaystyle{\\mathrm{Tr}}_{L/\\mathbb{Q}}(b_1\\alpha_1^j+\\cdots+b_k\\alpha_k^j) \\in \\mathbb{Z}$ for all $j = 1, 2, \\ldots, d+d[\\log_2(nd)]+1$, then $\\alpha_1$ is an algebraic integer.} We then prove a general result for the infinite version as follows. {\\it Suppose ${\\mathrm{Tr}}_{L/\\mathbb{Q}}(\\lambda_i\\alpha_i^a) \\ne 0$ for all integers $a\\in \\{0, 1, 2, \\ldots, h-1\\}$ and for all integers $i=1,\\ldots,k$. If ${\\mathrm{Tr}}_{L/\\mathbb{Q}}(\\lambda_1\\alpha_1^n+\\cdots +\\lambda_k \\alpha^n_k)\\in\\mathbb{Z}$ holds true for infinitely many natural numbers $n$, then each $\\alpha_i$ is an algebraic integer for all $i=1,\\ldots,k$. } Here the extra assumption is a necessary condition for $k > 1$ (See Remark 1.3). We also prove a Diophantine result which states: {\\it For a given rational number $p/q$, there are at most finitely many natural numbers $n$ such that ${\\mathrm{Tr}}_{L/\\mathbb{Q}}(\\lambda_1\\alpha_1^n+\\cdots +\\lambda_k \\alpha^n_k) = p/q$.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-09T13:26:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J87",
      "11S99"
    ],
    "keywords": [
      "linear combination",
      "natural numbers",
      "algebraic integer",
      "nonzero algebraic numbers",
      "diophantine result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}