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 $f\left(X\right)-f\left(Y\right)$ over a...

A number field $K$, with ring of integers ${𝒪}_K$, is said to be a Pólya field if the ${𝒪}_K$-algebra formed by the integer-valued polynomials on ${𝒪}_K$ admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field $K$ is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of $K$ in a Pólya field. We give a positive answer to this embedding problem by showing that...

A number field $K$, with ring of integers ${𝒪}_K$, is said to be a Pólya field when the ${𝒪}_K$-algebra formed by the integer-valued polynomials on ${𝒪}_K$ admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when $K$ is not a Pólya field, we are interested in the embedding of $K$ in a Pólya field. We study here two notions which can be considered as measures...

Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form $d=x(Cy+z)\frac{\partial }{\partial x}+y(Az+x)\frac{\partial }{\partial y}+z(Bx+y)\frac{\partial }{\partial z}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x₀,x₁,...,x_{k-1}$ of distinct elements of R is called a cycle of f if $f\left(x_i\right)=x_{i+1}$ for i=0,1,...,k-2 and $f\left(x_{k-1}\right)=x₀$. The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number $7^{7·2^N}$, depending only on the degree N of K. In this note we consider...

All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective. SISI rings include Noetherian rings, Morita rings and almost maximal valuation rings ([V1]). In [F3] we raised the question of whether a polynomial ring R[x] over a SISI ring R is again SISI. In this paper we show this is not the case.

Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q\left(m\right)$, where $Q$ is a polynomial with integral coefficients, then $P\left(x\right)=Q\left(R\right(x\left)\right)$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P\left(x\right)={\left(R\left(x\right)\right)}^q$ for some polynomial $R\left(x\right)$.