Related papers: Polynomial Index in Dedekind Rings

Let $p$ be a prime number and let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $N$ be a fully ramified, elementary abelian extension of $K$. Under a mild hypothesis on the extension $N/K$, we show that…

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating…

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\mathbb{F}_{q}[x]$ the ring of polynomials over $\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\mathbb{F}_{q}[x]$ and $\Phi(k)$ the Euler function in…

Let $a$ be an integer and $p$ a prime so that $f(x)=x^p-a$ is irreducible. Write $f^n(x)$ to indicate the $n$-fold composition of $f(x)$ with itself. We study the monogenicity of number fields defined by roots of $f^n(x)$ and give necessary…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

In this paper, the notions of nonnil-injective modules and nonnil-FP-injective modules are introduced and studied. Especially, we show that a $\phi$-ring $R$ is an integral domain if and only if any nonnil-injective (resp.,…

Let $L/K$ be a finite Galois extension of $p$-adic fields with group $G$. It is well-known that $\mathcal{O}_L$ contains a free $\mathcal{O}_K[G]$-submodule of finite index. We study the minimal index of such a free submodule, and determine…

Let $\psi: A \to F\{\tau\}$ be a Drinfeld $A$-module over $F$ of rank 2 and without complex multiplication, where $A = {\mathbb{F}}_q[T]$, $F = {\mathbb{F}}_q(T)$, and $q$ is an odd prime power. For a prime $\mathfrak{p} = p A$ of $A$ of…

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…

Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…

We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…

Let $X$ be a matrix with entries in a polynomial ring over an algebraically closed field $K$. We prove that, if the entries of $X$ outside some $(t \times t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of the…

Let $S=\mathbb{K}[x_1,\ldots, x_n]$ be the polynomial ring over a field $\mathbb{K}$ and $\mathfrak{m}= (x_1, \ldots, x_n)$ be the irredundant maximal ideal of $S$. For an ideal $I \subset S$, let $\mathrm{sat}(I)$ be the minimum number $k$…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both…

We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq 3$. We…

Let $k$ be an arbitrary field, $P = P_k^{m_1} \times_k \cdots \times_k P_k^{m_p}$ be a multiprojective space over $k$, and $X \subseteq P$ be a closed subscheme of $P$. We provide necessary and sufficient conditions for the positivity of…

As an extension of the classical irreducibility result of Dumas, a factorization result for polynomials over any valued field with a Krull valuation of arbitrary rank is proved. Further, a lower degree factor bound on factors of a given…

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in…