A derivation of Dickson polynomials using the Cayley-Hamilton theorem

2024/06/11 by Pain, Jean-Christophe
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2406.07322

Abstract

In this note, the first-order Dickson polynomials are introduced through a particular case of the expression of the trace of the n^th power of a matrix in terms of powers of the trace and determinant of the matrix itself. The technique relies on the Cayley-Hamilton theorem and its application to the derivation of formulas due to Carlitz and to second-order Dickson polynomials is straightforward. Finally, generalization of Dickson polynomials over finite fields and multivariate Dickson polynomials are evoked as potential avenues of investigation in the same framework.

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