A perfect polynomial over is a polynomial that equals the sum of all its divisors. If then we say that is odd. In this paper we show the non-existence of odd perfect polynomials with either three prime divisors or with at most nine prime divisors provided that all exponents are equal to
In this paper, we study the properties of the sequence of polynomials given by , for , where is non-constant and the characteristic of is . This complements some results from R. Euler, L.H. Gallardo: On explicit formulae and linear recurrent sequences, Acta Math. Univ. Comenianae, 80 (2011) 213-219.
Explicit monoid structure is provided for the class of canonical subfield preserving polynomials over finite fields. Some classical results and asymptotic estimates will follow as corollaries.
We estimate the number of possible degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is the number of primitive polynomials and ψ2 (n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large...