{
  "id": "2101.12180",
  "version": "v1",
  "published": "2021-01-28T18:40:29.000Z",
  "updated": "2021-01-28T18:40:29.000Z",
  "title": "Properties of solutions to Pell's equation over the polynomial ring",
  "authors": [
    "Nikoleta Kalaydzhieva"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\\sqrt{D}$ is the solution to Pell's equation for $D$. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions $(u_n,v_n)_{n\\in\\Zee}$. Our object of interest is the polynomial version of Pell's equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of $v_n(t)$. In particular, we show that over the complex polynomials, there are only finitely many values of $n$ for which $v_n(t)$ has a repeated root. Restricting our analysis to $\\Qee[t]$, we give an upper bound on the number of \"new\" factors of $v_n(t)$ of degree at most $N$. Furthermore, we show that all \"new\" linear rational factors of $v_n(t)$ can be found when $n\\leq 3$, and all \"new\" quadratic rational factors when $n\\leq 6$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T18:40:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pells equation",
      "polynomial ring",
      "properties",
      "linear rational factors",
      "quadratic rational factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}