On the Rank of Appearance and the Number of Zeros of the Lucas Sequences over $$ {\mathbb{F}_{q}} $$

Let D n (P,Q) respectively E n (P,Q) denote the Dickson polynomial of the first, respectively second kind of degree n in the indeterminate P and with parameter Q. It is well known that these polynomials are periodic over $$ {\mathbb{F}_{q}} $$ , the finite field consisting of q elements. That is, there exist some positive integers t 1,t 2 such that $$ {D_{{n + {t_{1}}}}}(P,Q){\text{ }} = {\text{ }}{D_{n}}(P,Q) $$ , respectively $$ {E_{{n + {t_{2}}}}}(P,Q) = {E_{n}}(P,Q) $$ for all n ≥ 0. In this paper we will be dealing with the periodicity of the occurrence of 0. This special type of periodicity is known as the rank of appearance of the corresponding Lucas sequence (respectively Dickson polynomial). For any possible value of the rank of appearance we will give the exact number of parameters with this specific rank over $$ {\mathbb{F}_{q}} $$ . We will firstly adopt the ideas of [12] in keeping Q = Q 0 fixed and considering D n and E n as polynomials in P. Conversely then we will also be dealing with the opposite case, where we keep P = P 0 fixed and regard Q as the indeterminate. Our main results will extend previous results in [8] and [9] concerning parameters with the same rank of appearance over $$ {\mathbb{F}_{p}} $$ , p an odd prime. We will show that similar results can be obtained over any finite field $$ {\mathbb{F}_{q}} $$ . In the second part of the paper we show how this number of parameters with the same rank can efficiently be utilized into establishing the number of zeros of the Dickson polynomials. That is, for any fixed Q = Q 0 we will establish the number of zeros P of both of the Dickson polynomials D n (P,Q 0) and E n P,Q 0) over $$ {\mathbb{F}_{q}} $$ and give distinct formulas which are specified by certain attributes of the parameter Q 0 Again, we then also reverse the role of P and Q in that we keep P = P 0 fixed, and regarding D n P 0, Q) and E n (P 0, Q) as polynomials in the variable Q, we will establish the number of zeros Q over $$ {\mathbb{F}_{q}} $$ . Most interestingly, in this case, the formulas will not depend on P 0.