相关论文: Polynomials over Ring of Integers of Global Fields…

Let~$E$ be a Hilbertian field of characteristic~$0$. R.W.K. Odoni conjectured that for every positive integer~$n$ there exists a polynomial~$f\in E[X]$ of degree~$n$ such that each iterate~$f^{\circ{k}}$ of~$f$ is irreducible and the Galois…

This paper proves a stability result for a variation of the Erd\H{o}s-Ko-Rado theorem in the context of polynomials over finite fields. Let $\mathcal F$ be a family of polynomials of degree at most $k \geq 3$ in $\mathbb F_q[X]$. Call…

Consider the conjugation action of the general linear group $\operatorname{GL}_{2}(K)$ on the polynomial ring $K[X_{2 \times 2}]$. When $K$ is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the…

Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the field of fractions $\K$. Denote the ring of polynomials in $n$ variables over $\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by…

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic…

Let $f(x) = \sum\limits _{i=0}^{n} a_i x^i $ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0 < a_0 \leq a_{1} \leq \cdots \leq a_{k-1} < a_{k} < a_{k+1} \leq \cdots \leq a_n$ for some $k$,…

Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta$ be a root of $f(x)$, set $K= \mathbb{Q}(\theta)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted…

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$,…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…

This is essentially an erratum, with some example to indicate inconsistencies. Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$. The Complete Intersection conjecture states that, for any ideal $I$ in $A$,…

We prove an effective form of Hilbert's irreducibility theorem for polynomials over a global field $K$. More precisely, we give effective bounds for the number of specializations $t\in \mathcal{O}_K$ that do not preserve the irreducibility…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

Given an integral domain $D$ with quotient field $K$, the ring of integer-valued polynomials on D is the subring $\{f (X) \in K[X]: f(D) \subset D\}$ of the polynomial ring $K[X]$. Using the related tools of $t$-closure and associated…

Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u…

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers…

This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting…