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arXiv · 2103.09119

Residue sums of Dickson polynomials over finite fields

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

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Thomas Brazelton, Joshua Harrington, Matthew Litman, Tony W. H. Wong. 2021-03-16. Residue sums of Dickson polynomials over finite fields. https://doi.org/10.1016/j.jnt.2024.04.016

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