Enumeration of Special Sets of Polynomials over Finite Fields

In this paper we consider squarefree polynomials over finite fields whose gcd with their reciprocal and Frobenius conjugate polynomial is trivial, respectively. Our focus is on the enumeration of these special sets of polynomials, in particular, we give the number of squarefree palindromes. These investigations make it possible to estimate the efficiency of the prefactorization of polynomials over finite fields based on such gcd calculations