{
  "id": "1304.4635",
  "version": "v1",
  "published": "2013-04-16T22:13:16.000Z",
  "updated": "2013-04-16T22:13:16.000Z",
  "title": "Patterns In The Coefficients Of Powers Of Polynomials Over A Finite Field",
  "authors": [
    "Kevin Garbe"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We examine the behavior of the coefficients of powers of polynomials over a finite field of prime order. Extending the work of Allouche-Berthe, 1997, we study a(n), the number of occurring strings of length n among coefficients of any power of a polynomial f reduced modulo a prime p. The sequence of line complexity a(n) is p-regular in the sense of Allouche-Shalit. For f=1+x and general p, we derive a recursion relation for a(n) then find a new formula for the generating function for a(n). We use the generating function to compute the asymptotics of a(n)/n^2 as n approaches infinity, which is an explicitly computable piecewise quadratic in x with n= [p^m/x] and x is a real number between 1/p and 1. Analyzing other cases, we form a conjecture about the generating function for general a(n). We examine the matrix B associated with f and p used to compute the count of a coefficient, which applies to the theory of linear cellular automata and fractals. For p=2 and polynomials of small degree we compute the largest positive eigenvalue, \\lambda, of B, related to the fractal dimension d of the corresponding fractal by d= \\log_2(\\lambda). We find proofs and make a number of conjectures for some bounds on \\lambda, and upper bounds on its degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-16T22:13:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite field",
      "coefficient",
      "polynomial",
      "generating function",
      "linear cellular automata"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.4635G"
    }
  }
}