A derivation of Dickson polynomials using the Cayley-Hamilton theorem

TL;DR

This paper derives first-order Dickson polynomials using the Cayley-Hamilton theorem, providing a matrix-based approach that connects polynomial expressions with matrix trace and determinant, and discusses potential generalizations.

Contribution

It introduces a novel derivation of Dickson polynomials via the Cayley-Hamilton theorem, linking matrix theory with polynomial construction.

Findings

Derivation of first-order Dickson polynomials from matrix trace expressions

Application of Cayley-Hamilton theorem to polynomial formulas

Discussion of potential generalizations to finite fields and multivariate cases

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.