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Improvements on the Cantor-Zassenhaus factorization algorithm

Michele Elia, Davide Schipani (2015)

Mathematica Bohemica

The paper presents a careful analysis of the Cantor-Zassenhaus polynomial factorization algorithm, thus obtaining tight bounds on the performances, and proposing useful improvements. In particular, a new simplified version of this algorithm is described, which entails a lower computational cost. The key point is to use linear test polynomials, which not only reduce the computational burden, but can also provide good estimates and deterministic bounds of the number of operations needed for factoring....

Irreducibility of the iterates of a quadratic polynomial over a field

Mohamed Ayad, Donald L. McQuillan (2000)

Acta Arithmetica

1. Introduction. Let K be a field of characteristic p ≥ 0 and let f(X) be a polynomial of degree at least two with coefficients in K. We set f₁(X) = f(X) and define f r + 1 ( X ) = f ( f r ( X ) ) for all r ≥ 1. Following R. W. K. Odoni [7], we say that f is stable over K if f r ( X ) is irreducible over K for every r ≥ 1. In [6] the same author proved that the polynomial f(X) = X² - X + 1 is stable over ℚ. He wrote in [7] that the proof given there is quite difficult and it would be of interest to have an elementary proof. In the sequel...

Lifting solutions over Galois rings.

Javier Gómez-Calderón (1990)

Extracta Mathematicae

In this note we generalize some results from finite fields to Galois rings which are finite extensions of the ring Zpm of integers modulo pm where p is a prime and m ≥ 1.

Linear recurrence sequences without zeros

Artūras Dubickas, Aivaras Novikas (2014)

Czechoslovak Mathematical Journal

Let a d - 1 , ⋯ , a 0 ∈ ℤ , where d ∈ ℕ and a 0 ≠ 0 , and let X = ( x n ) n = 1 ∞ be a sequence of integers given by the linear recurrence x n + d = a d - 1 x n + d - 1 + ⋯ + a 0 x n for n = 1 , 2 , 3 , ⋯ . We show that there are a prime number p and d integers x 1 , ⋯ , x d such that no element of the sequence X = ( x n ) n = 1 ∞ defined by the above linear recurrence is divisible by p . Furthermore, for any nonnegative integer s there is a prime number p ≥ 3 and d integers x 1 , ⋯ , x d such that every element of the sequence X = ( x n ) n = 1 ∞ defined as above modulo p belongs to the set { s + 1 , s + 2 , ⋯ , p - s - 1 } .

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