Given a monic degree $N$ polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ and a non-negative integer $\ell$, we may form a new monic degree $N$ polynomial $f_{\ell }\left(x\right)\in \mathbb{Z}\left[x\right]$ by raising each root of $f$ to the $\ell$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 $(j,k)\mapsto Res(f_j,f_k)mod{p}^m$. 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.

Given β > 1, let Tβ$\begin{array}{ccc}{\mathit{T}}_{\mathit{\beta }}\mathrm{:}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}& \mathrm{\to }& \mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}\\ \mathit{x}& \mathrm{\to }& \mathit{\beta x}\mathrm{-}\mathrm{\lfloor }\mathit{\beta x}\mathrm{\rfloor }\mathit{.}\end{array}$The iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of Tβ, where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion$\begin{array}{ccc}\mathit{x}\mathrm{=}\sum _{\mathit{i}\mathrm{\ge }\mathrm{0}}{\mathit{e}}_{\mathit{i}}{\mathit{\beta }}^{\mathrm{-}\mathit{i}}\mathit{,}& & \end{array}$where each ei...

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.

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).

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 $𝐙\left[\frac{1}{N}\right]$ and shall describe polynomial cycles in the case when $N$ is either odd or twice a prime.

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.

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...

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.

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.