Gary Mullen,
Penn State University
Dickson Polynomials over Finite Fields: two different perspectives
Abstract:
If a is in F_q, the finite field of order q, the Dickson polynomial of degree n and parameter a is defined by D_n(x,a) = \sum _{i=0} ^{\lfloor n/2 \rfloor} \frac {n} {n-i} {n-i \choose i} (-a)^i x^{n-2i}.
Dickson polynomials over finite fields have many very interesting properties. I will first briefly discuss some of these properties, hoping to generate further interest in these very fascinating polynomials.
Some of the properties of Dickson polynomials are related to questions of permutations of finite fields. In previous work, the parameter a has been fixed and the variable x then runs through the field F_q.
In some current work with James Sellers (Penn State) and Joe Yucas (Southern Illinois). we reverse these roles, and fix x in F_q, and then allow a to run through the elements of the field F_q. It appears that once again, we have an interesting, though far from understood, class of polynomials.