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Residue sums of Dickson polynomials over finite fields

Thomas Brazelton, Joshua Harrington, Matthew Litman, Tony W. H. Wong

Abstract

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

Residue sums of Dickson polynomials over finite fields

Abstract

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime . The sum over the distinct residues can sometimes be computed independent of the prime ; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

Paper Structure

This paper contains 10 sections, 27 theorems, 71 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dickson polynomials
  4. Hyperbolic, elliptic, and parabolic elements
  5. Evaluation of the residue sum
  6. Summing over the hyperbolic and elliptic images
  7. Overlaps in the non-residue case
  8. Overlaps: the residue case
  9. Examples
  10. Further Directions and Conclusion

Key Result

Theorem 1.2

Let $a$ be an integer, $n$ be a nonnegative integer, and $q$ be an odd prime power such that $a\neq 0 \in \mathbb{F}_q$. Let $d=\gcd(n,q-1)$ and $\delta = \gcd(n,q+1)$, and let $r$ be the highest power of 2 dividing $q^2-1$. Then the sum of the elements in the image of the Dickson polynomials isWhen

Figures (3)

  • Figure 1: Possible values for $S_7(L_n)$.
  • Figure 2: Possible values for $S_{29}(L_n)$.
  • Figure 3: Investigation of $S_7(F_{2n-1})$ for $1\le n\le 40$.

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.6
  • ...and 38 more