{
  "id": "2111.13745",
  "version": "v3",
  "published": "2021-11-26T20:58:59.000Z",
  "updated": "2023-05-22T21:49:03.000Z",
  "title": "Combinatorial Relationship Between Finite Fields and Fixed Points of Functions Going Up and Down",
  "authors": [
    "Emerson León",
    "Julián Pulido"
  ],
  "comment": "17 pages, 6 figures",
  "categories": [
    "math.CO",
    "math.DS",
    "math.NT"
  ],
  "abstract": "We explore a combinatorial bijection between two seemingly unrelated topics: the roots of irreducible polynomials of degree $m$ over a finite field $F_p$ for a prime number $p$ and the number of points that are periodic of order $m$ for a continuous piece-wise linear function $g_p:[0,1]\\rightarrow[0,1]$ that \\emph{goes up and down $p$ times} with slope $\\pm 1/p$. We provide a bijection between $F_{p^n}$ and the fixed points of $g^n_p$ that naturally relates some of the structure in both worlds. Also we extend our result to other families of continuous functions that goes up and down $p$ times, in particular to Chebyshev polynomials, where we get a better understanding of its fixed points. A generalization for other piece-wise linear functions that are not necessarily continuous is also provided.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2023-05-22T21:49:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fixed points",
      "finite field",
      "combinatorial relationship",
      "continuous piece-wise linear function",
      "chebyshev polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}