On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials

Borissov, Yuri, Ho Lee, Moon, and Nikova, Svetla. "On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials." Serdica Journal of Computing 2.3 (2008): 239-248. <http://eudml.org/doc/11464>.

@article{Borissov2008,
abstract = {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 primes pi . We also describe an infinite series of values ns such that θ(ns ) is strictly less than 1/2.},
author = {Borissov, Yuri, Ho Lee, Moon, Nikova, Svetla},
journal = {Serdica Journal of Computing},
keywords = {Finite Fields; Primitive and Irreducible Polynomials; irreducible polynomials; primitive polynomials; finite fields},
language = {eng},
number = {3},
pages = {239-248},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials},
url = {http://eudml.org/doc/11464},
volume = {2},
year = {2008},
}

TY - JOUR
AU - Borissov, Yuri
AU - Ho Lee, Moon
AU - Nikova, Svetla
TI - On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials
JO - Serdica Journal of Computing
PY - 2008
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 2
IS - 3
SP - 239
EP - 248
AB - 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 primes pi . We also describe an infinite series of values ns such that θ(ns ) is strictly less than 1/2.
LA - eng
KW - Finite Fields; Primitive and Irreducible Polynomials; irreducible polynomials; primitive polynomials; finite fields
UR - http://eudml.org/doc/11464
ER -