Related papers: An irreducibility criterion for power series
This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling…
In this paper we study the index of reducibility of powers of a standard parameter ideal. An explicit formula is proved for the extremely case. We apply the main result to compute Hilbert polynomials of socle ideals of standard parameter…
We generalize Carlitz' result on the number of self reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula hold for the number of irreducible polynomials obtained by a fixed quadratic…
Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…
We deal with the algebraicity of a Puiseux series in terms of the properties of its coefficients. We show that the algebraicity of a Puiseux series for given bounded degree is determined by a finite number of explicit polynomial formulae.…
Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most…
Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying…
We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator,…
In this paper the author considers a particular type of polynomials with integer coefficients, consisting of a perfect power and two norm forms of abelian number fields with coprime discriminants. It is shown that such a polynomial…
We prove the following theorem: Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive dimensional varieties of general type, with $B$ irreducible. Then there exists an integer $n>0$, a positive dimensional variety of general…
We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is…
We consider the question of certifying that a polynomial in ${\mathbb Z}[x]$ or ${\mathbb Q}[x]$ is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv.~that a…
Recently it was shown that it is undecidable whether a term rewrite system can be proved terminating by a polynomial interpretation in the natural numbers. In this paper we show that this is also the case when restricting the…
In this paper, we prove an inequality regarding the differential polynomial. This improves some recent results.
In this short note, a general result concerning the positivity, under some conditions, of the coefficients of a power series is proved. This allows us to answer positively a question raised by Guo (2010) about the sign of the coefficients…
A sequence of coefficients that appeared in the evaluation of a rational integral has been shown to be unimodal. An alternative proof is presented.
We study certain kind of polynomials associated with Lissajous curves, called Chebyshev-Lissajous polynomials. We investigate their irreducibilities over the real numbers and complex numbers, thus comfirming two conjectures proposed by…
A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial…
Jakhar shown that for $f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdot+ a_0$ ($a_0\neq 0$) is a polynomial with rational coefficients, if there exists a prime integer $p$ satisfying $\nu_p(a_n)=0$ and $n\nu_p(a_i)\ge (n-i)\nu_p(a_0)> 0$ for every $0\le…
In this article we study asymptotic behavior of the probability that a random monic polynomial with integer coefficients is irreducible over the integers. We consider the cases where the coefficients grow together with the degree of the…