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Parametrization of integral values of polynomials

Giulio Peruginelli (2010)

Actes des rencontres du CIRM

We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is 2 and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial f ( X ) - f ( Y ) over a...

Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

Given a monic degree N polynomial f ( x ) ∈ ℤ [ x ] and a non-negative integer ℓ , we may form a new monic degree N polynomial f ℓ ( x ) ∈ ℤ [ x ] by raising each root of f to the ℓ th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) ↦ Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

Polynômes de Lagrange sur les entiers d'un corps quadratique imaginaire

M. Ably, M. M'Zari (1998)

Journal de théorie des nombres de Bordeaux

L'objet de ce texte est de donner une estimation arithmétique des valeurs prises par les polynômes de Lagrange sur les entiers d'un corps quadratique imaginaire en des points de ce corps. Ces polynômes interviennent dans l'étude des fonctions entières arithmétiques et dans les minorations de formes linéaires de Logarithmes.

Polynomial Automorphisms Over Finite Fields

Maubach, Stefan (2001)

Serdica Mathematical Journal

It is shown that the invertible polynomial maps over a finite field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1 it is shown that the tame subgroup of the invertible polynomial maps gives only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if #S = q^(n−1).

Polynomial cycles in certain rings of rationals

Władysław Narkiewicz (2002)

Journal de théorie des nombres de Bordeaux

It is shown that the methods established in [HKN3] can be effectively used to study polynomial cycles in certain rings. We shall consider the rings 𝐙 [ 1 N ] and shall describe polynomial cycles in the case when N is either odd or twice a prime.

Polynomial orbits in finite commutative rings

Petra Konečná (2006)

Czechoslovak Mathematical Journal

Let R be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.

Poznámka o celočíselných polynomech, jejichž hodnoty jsou dělitelné číslem n !

Vlastimil Dlab (2013)

Učitel matematiky

Článek si dává za cíl ukázat, že z kanonických polynomů Dn(x) lze pomocí určitých lineárních kombinací vytvořit všechny polynomy, které jsou dělitelné n!. Autor formuluje větu o dělitelnosti těchto polynomů n!. Z této věty pak vyplývá celá řada tvrzení, z kterých uvádí pouze prvních šest. V každém tvrzení nalezne polynom a postupně tvrdí, že první je dělitelný 2, další 6, další 24, další číslem 120, další 720 a poslední 5040 pro celočíselné koeficienty. Vzhledem k těmto tvrzením formuluje obecné...

Prime factors of values of polynomials

J. Browkin, A. Schinzel (2011)

Colloquium Mathematicae

We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.

Prime rational functions

Omar Kihel, Jesse Larone (2015)

Acta Arithmetica

Let f(x) be a complex rational function. We study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.

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