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Counting monic irreducible polynomials P in 𝔽 q [ X ] for which order of X ( mod P ) is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

Hasse showed the existence and computed the Dirichlet density of the set of primes p for which the order of 2 ( mod p ) is odd; it is 7 / 24 . Here we mimic successfully Hasse’s method to compute the density δ q of monic irreducibles P in 𝔽 q [ X ] for which the order of X ( mod P ) is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the δ p ’s as p varies through all rational primes.

Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

Let f = a x + b x q + x 2 q - 1 ∈ q [ x ] . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of q ² . This result allows us to solve a related problem: Let g n , q ∈ p [ x ] (n ≥ 0, p = c h a r q ) be the polynomial defined by the functional equation ∑ c ∈ q ( x + c ) n = g n , q ( x q - x ) . We determine all n of the form n = q α - q β - 1 , α > β ≥ 0, for which g n , q is a permutation polynomial of q ² .

Diagonalization and rationalization of algebraic Laurent series

Boris Adamczewski, Jason P. Bell (2013)

Annales scientifiques de l'École Normale Supérieure

We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime p the reduction modulo p of the diagonal of a multivariate algebraic power series f with integer coefficients is an algebraic power series of degree at most p A and height at most A p A , where A is an effective constant that only depends on...

Dickson curves.

Gomez-Calderon, Javier (2006)

International Journal of Mathematics and Mathematical Sciences

Dickson Polynomials that are Permutations

Cipu, Mihai (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 11T06, 13P10.A theorem of S.D. Cohen gives a characterization for Dickson polynomials of the second kind that permutes the elements of a finite field of cardinality the square of the characteristic. Here, a different proof is presented for this result.Research supported by the CERES program of the Ministry of Education, Research and Youth, contract nr. 39/2002.

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