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1. Counting irreducible polynomials over finite fields

Creator:
Wang, Qichun and Kan, Haibin
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
finite fields, distribution of irreducible polynomials, and residue
Language:
English
Description:
In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: \[ \pi (x)= \frac q{q - 1}\frac x{{\log _q x}}+ \frac q{(q - 1)^2}\frac x{{\log _q^2 x}}+O\Bigl (\frac {x}{{\log _q^3 x}}\Bigr ),\quad x=q^n\rightarrow \infty \] where $\pi (x)$ denotes the number of monic irreducible polynomials in $F_q [t]$ with norm $ \le x$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Quasi-permutation polynomials

Creator:
Laohakosol, Vichian and Janphaisaeng, Suphawan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
finite fields and permutation polynomials
Language:
English
Description:
A quasi-permutation polynomial is a polynomial which is a bijection from one subset of a finite field onto another with the same number of elements. This is a natural generalization of the familiar permutation polynomials. Basic properties of quasi-permutation polynomials are derived. General criteria for a quasi-permutation polynomial extending the well-known Hermite's criterion for permutation polynomials as well as a number of other criteria depending on the permuted domain and range are established. Different types of quasi-permutation polynomials and the problem of counting quasi-permutation polynomials of fixed degree are investigated.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public