A derivation of Dickson polynomials using the Cayley-Hamilton theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913389642514432 |
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| author | Pain, Jean-Christophe |
| author_facet | Pain, Jean-Christophe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_07322 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A derivation of Dickson polynomials using the Cayley-Hamilton theorem Pain, Jean-Christophe Number Theory 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. |
| title | A derivation of Dickson polynomials using the Cayley-Hamilton theorem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.07322 |