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A trigonometric approach for Dickson polynomials over fields of characteristic two. (English) Zbl 1459.11222

The authors study Dickson polynomials \(D_i(x,1)\) and \(E_i(x,1)\) over finite fields \(\mathbb{F}_{2^n}\), \(n\ge 1\). In Section 2, they develop trigonometry over fields of characteristic two to an extent sufficient to prove that the polynomials obtained by the well known recurrence coincide with those based on a definition imitating the trigonometric approach from the classical context. Next, the focus is on periodicity, permutation and involution properties of such polynomials. The results for \(D_i\) parallel the classical characterization. Without surprise, the study of Dickson polynomials of the second kind is more intricate and produces less results. In Section 4, the authors succeed in establishing several results on the fixed points of \(E_i\), as well as some sufficient conditions regarding the possibility of being permutation or involution of \(\mathbb{F}_{2^n}\).

MSC:

11T06 Polynomials over finite fields
11T22 Cyclotomy
12E20 Finite fields (field-theoretic aspects)
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