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Jitender Singh

Publications and source records attributed to Jitender Singh.

At least 19 recordsLinked to original sources
arXiv2025

On irreducible factors of polynomials over integers

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their higher order formal derivatives at sufficiently large integer arguments. If a lower bound for the minimum possible degree of a factor of such a polynomial is known a priori, then the integer argument becomes significantly smaller, which makes the underlying factorization result easier to apply. A result on explicit lower degree factor bound for the classes of polynomials considered in this paper is also proved via Newton polygons.

math.NT↗
arXiv2025

A generalized Dumas irreducibility criterion

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, among other results, a lower degree factor bound on factors of a given polynomial over a valued field with a Krull valuation is proved. These factorization results not only unify several known irreducibility results for polynomials over the said domains but also provide us sharp bounds on degrees of irreducible factors of the underlying polynomials.

math.NT↗
arXiv2025

A generalization of Dumas irreducibility criterion

Using Newton polygons, a key factorization result for polynomials over discrete valuation domains is proved, which in particular yields new irreducibility criteria including a generalization of the classical irreducibility criterion of Dumas.

math.NT↗
arXiv2025

Some factorization results for formal power series

In this paper, we obtain some factorization results on formal power series over principle ideal domains with sharp bounds on number of irreducible factors. These factorization results correspondingly lead to irreducibility criteria for formal power series. The information about prime factorization of the constant term up to a unit and that of some higher order terms is utilized for the purpose. Further, using theory of Newton polygons for power series, we extend the classical Dumas irreducibility criterion to formal power series over discrete valuation domains, which in particular, yields several irreducibility criteria.

math.NT↗
arXiv2024

Prime numbers and factorization of polynomials

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we use the information about prime factorization of the values taken by such polynomials at sufficiently large integer arguments along with the information about their root location in the complex plane. Further, these techniques are extended to bivariate polynomials over arbitrary fields using non-Archimedean absolute values, yielding extensions of the irreducibility results of M. Ram Murty and S. Weintraub to bivariate polynomials.

math.NT↗
arXiv2024

Some factorization results for bivariate polynomials

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the degrees of the coefficients of $f$, and on information on the factorization of the constant term and of the leading coefficient of $f$, viewed as a polynomial in $y$ with coefficients in $\mathbb{K}[x]$. In particular, we provide a generalization of the bivariate version of Perron's irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of $f$ in an algebraic closure of $\mathbb{K}(x)$.

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arXiv2023

Irreducibility via location of zeros

In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials are devised by imposing Perron--type sufficiency conditions on their coefficients.

math.NT↗
arXiv2023

Some factorization results on polynomials having integer coefficients

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing some sufficiency conditions on their coefficients along with some conditions on the prime factorization of constant term or the leading coefficient of the underlying polynomials in conjunction with the information about their root location.

math.NT↗
arXiv2023

On a factorization result of Ştefănescu -- II

Ştefănescu proved an elegant factorization result for polynomials over discrete valuation domains [CASC'2014, Lecture Notes in Computer Science, Ed. by V. Gerdt, W. Koepf, W. Mayr, and E. Vorozhtsov, Springer, Berlin, {Vol. \textbf{8660}}, pp. 460--471, 2014.] In this paper, a generalization of Ştefănescu's result is proved to cover a larger class of polynomials over discrete valuation domains. Such results are useful in devising algorithms for polynomial factorization.

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arXiv2022

Another irreducibility criterion

Let $f=a_0+ a_{1}x+\cdots+a_m x^m\in \Bbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $α$ such that $|a_m| α^m>|a_0|+|a_1|α+\cdots+|a_{m-1}|α^{m-1}$. We prove that if there exist natural numbers $n$ and $d$ satisfying $n\geq α+ d$ for which either $|f(n)|/d$ is a prime, or $|f(n)|/d$ is a prime-power coprime to $|f'(n)|$, then $f$ is irreducible in $\mathbb{Z}[x]$.

math.NT↗
arXiv2022

Round and sleek subspaces of linear metric spaces and metric spaces

In the recent work [Metrically round and sleek metric spaces, \emph{The Journal of Analysis} (2022), pp 1--17], the authors proved some results on metrically round and sleek linear metric spaces and metric spaces. In continuation, the present article discusses more results on such spaces along with identification of round and sleek subsets of linear metric spaces and metric spaces in the subspace topology.

math.MG↗
arXiv2022

Shooting method for solving two-point boundary value problems in ODEs numerically

Boundary value problems in ODEs arise in modelling many physical situations from microscale to mega scale. Such two-point boundary value problems (BVPs) are complex and often possess no analytical closed form solutions. So, one has to rely on approximating the actual solution numerically to a desired accuracy. To approximate the solution numerically, several numerical methods are available in the literature. In this chapter, we explore on finding numerical solutions of two-point BVPs arising in higher order ODEs using the shooting technique. To solve linear BVPs, the shooting technique is derived as an application of linear algebra. We then describe the nonlinear shooting technique using Newton-Kantorovich theorem in dimension n>1. In the one-dimensional case, Newton-Raphson iterates have rapid convergence. This is not the case in higher dimensions. Nevertheless, we discuss a class of BVPs for which the rate of convergence of the underlying Newton iterates is rapid. Some explicit examples are discussed to demonstrate the implementation of the present numerical scheme.

physics.flu-dyn↗
arXiv2022

Longitudinal and transverse modes of temperature modulated inclined layer convection

A parametric instability of an incompressible, viscous, and Boussinesq fluid layer bounded between two parallel planes is investigated numerically. The layer is assumed to be inclined at an angle with horizontal. The planes bounding the layer are subjected to a time-periodic heating. Above a threshold value, the temperature gradient across the layer leads to an instability of an initially quiescent state or a parallel flow, depending upon the angle of inclination. The Floquet analysis of the underlying system reveals that under modulation, the instability sets in as a convective roll pattern executing harmonic or subharmonic oscillations, depending upon the modulation, the angle of inclination, and Prandtl number of the fluid. Under modulation, the value of the angle of inclination for the codimension-2 point is found to be a nonconstant function of the amplitude and the frequency of modulation. Further, the instability response in the fluid layer as a longitudinal mode is always harmonic whereas the instability response as a transverse mode is harmonic, or subharmonic, or bicritical depending upon the modulation. The temperature modulation offers a good control of time-periodic heat and mass transfer in the inclined layer convection.

physics.flu-dyn↗
arXiv2021

On a factorization result of Ştefănescu

In this article, some factorization properties of polynomials over discrete valuation domains are elucidated. These properties along with the notion of Newton index of a polynomial leads to a generalization of the main result proved by Ştefănescu [`On the factorization of polynomials over discrete valuation domains', \emph{Versita} \textbf{22}:1 (2014), 273--280].

math.NT↗
arXiv2021

Metrically Round and Sleek Metric Spaces

A round metric space is the one in which closure of each open ball is the corresponding closed ball. By a sleek metric space, we mean a metric space in which interior of each closed ball is the corresponding open ball. In this, article we establish some results on round metric spaces and sleek metric spaces.

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