Monic Polynomials Research Articles

Proper Improvement of Well-Known Numerical Radius Inequalities and Their Applications

New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space \({\mathcal {H}}\) are given. In particular, it is established that if T is a bounded linear operator on a Hilbert space \({\mathcal {H}}\) then $$\begin{aligned} w^2(T)\le \min _{0\le \alpha \le 1} \left\| \alpha T^*T +(1-\alpha )TT^* \right\| , \end{aligned}$$where w(T) is the numerical radius of T. The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a complex monic polynomial.

Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

We study the Smith forms of matrices of the form f(Cg) where f(t),g(t)∈R[t], where R is an elementary divisor domain and Cg is the companion matrix of the (monic) polynomial g(t). Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix f(Cg) to that of the matrix F(CG), where F,G are quotients of f(t),g(t) by some common divisor. This allows us to express the last non-zero determinantal divisor of f(Cg) as a resultant. A key tool is the observation that a matrix ring generated by Cg – the companion ring of g(t) – is isomorphic to the polynomial ring Qg=R[t]/<g(t)>. We relate several features of the Smith form of f(Cg) to the properties of the polynomial g(t) and the equivalence classes [f(t)]∈Qg. As an application we let f(t) be the Alexander polynomial of a torus knot and g(t)=tn−1, and calculate the Smith form of the circulant matrix f(Cg). By appealing to results concerning cyclic branched covers of knots and cyclically presented groups, this provides the homology of all Brieskorn manifolds M(r,s,n) where r,s are coprime.

On monogenity of certain pure number fields defined by xpr − m

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

Boundedness results for 2‐adic Galois images associated to hyperelliptic Jacobians

AbstractLet K be a number field, and let C be a hyperelliptic curve over K with Jacobian J. Suppose that C is defined by an equation of the form for some irreducible monic polynomial of discriminant Δ and some element . Our first main result says that if there is a prime of K dividing but not (2Δ), then the image of the natural 2‐adic Galois representation is open in and contains a certain congruence subgroup of depending on the maximal power of dividing . We also present and prove a variant of this result that applies when C is defined by an equation of the form for distinct elements . We then show that the hypothesis in the former statement holds for almost all and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.

Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Based on our observation, we find that this weight includes the symmetric constraint [Formula: see text]. Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree [Formula: see text] goes to infinity. Because of the effect of [Formula: see text] for varying [Formula: see text], the asymptotic behavior in a neighborhood of point [Formula: see text] is described in terms of the Airy function as [Formula: see text], but the Bessel function as [Formula: see text]. Due to symmetry, the similar local asymptotic behavior near the singular point [Formula: see text] can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

Spectral Transformations and Associated Linear Functionals of the First Kind

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

AbstractWe study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as have recently been studied for , and our main goal is to extend these results to all in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter approaches a breaking curve, by considering double scaling limits as approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points or some other points on the breaking curve.

Differential, difference, and asymptotic relations for Pollaczek–Jacobi type orthogonal polynomials and their Hankel determinants

AbstractIn this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek–Jacobi type weight urn:x-wiley:00222526:media:sapm12392:sapm12392-math-0001By using the ladder operator approach, we establish the second‐order difference equations satisfied by the recurrence coefficient and the sub‐leading coefficient of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of can be expressed in terms of a particular Painlevé V transcendent. The large asymptotic expansions of and are obtained by using Dyson's Coulomb fluid method together with the related difference equations. Furthermore, we study the associated Hankel determinant and show that a quantity , allied to the logarithmic derivative of , can be expressed in terms of the ‐function of a particular Painlevé V. The second‐order differential and difference equations for are also obtained. In the end, we derive the large asymptotics of and from their relations with and .

On integral bases and monogeneity of pure sextic number fields with non-squarefree coefficients

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f(x)=x6−m∈Z[x], it was assumed that the rational integer m≠∓1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of ZK; the ring of integers of K. In particular, we characterize when ZK=Z[α], that is when ZK is monogenic and generated by α. We give sufficient conditions on m, which warranty that K is not monogenic. We finish the paper by investigating the case, where m=e5 and e≠∓1 is a square free rational integer.

The integral moments and ratios of quadratic Dirichlet L-functions over monic irreducible polynomials in mathbb {F}_{q}[T

In this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions L(s,chi _{P}) where the character chi is defined by the Legendre symbol for polynomials in mathbb {F}_{q}[T] with mathbb {F}_{q} a finite field of odd cardinality, and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application, we also compute the formula for the one-level density for the zeros of these L-functions.

Cohen–Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields

Let (R,m) be a complete discrete valuation ring with the finite residue field R∕m= Fq. Given a monic polynomial P(t)∈R[t] whose reduction modulo m gives an irreducible polynomial P‾(t)∈Fq[t], we initiate an investigation of the distribution of coker(P(A)), where A∈Matn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n×n R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between Matn(R) and Matn(Fq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(Fq) with respect to the uniform distribution. Our results over Fq are about the distribution of the P‾-part of a random matrix A‾∈Matn(Fq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |AutZ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

Eigenvalue paths arising from matrix paths

It is known (see e.g. [2], [4], [5], [6]) that continuous variations in the entries of a complex square matrix induce continuous variations in its eigenvalues. If such a variation arises from one real parameter α∈[0,1], then the eigenvalues follow continuous paths in the complex plane as α shifts from 0 to 1. The intent here is to study the nature of these eigenpaths, including their behavior under small perturbations of the matrix variations, as well as the resulting eigenpairings of the matrices that occur at α=0 and α=1. We also give analogs of our results in the setting of monic polynomials.

Painlevé V for a Jacobi unitary ensemble with random singularities

In this paper, we focus on the relationship between the fifth Painlevé equation and a Jacobi weight perturbed with random singularities, w(z)=1−z2αe−tz2−k2,z,k∈[−1,1],α,t>0.By using the ladder operator approach, we obtain that an auxiliary quantity Rn(t), which is closely related to the recurrence coefficients of monic polynomials orthogonal with w(z), satisfies a particular Painlevé V equation.

The least common multiple of random sets in polynomial rings over finite fields

In this paper we investigate the distribution of degrees of the least common multiples of random subsets of monic polynomials of degree n in $${\mathbb {F}}_q[t]$$ . We consider two different probabilistic models and find the concentration of degrees around the corresponding expectations when $$q \rightarrow \infty $$ or $$n \rightarrow \infty $$ .

English

Let K be a number field defined by an irreducible polynomial F(X) ∈ ℤ[X] and ℤK its ring of integers. For every prime integer p, we give sufficient and necessary conditions on F(X) that guarantee the existence of exactly r prime ideals of ℤK lying above p, where $$\overline F \left( X \right)$$ factors into powers of r monic irreducible polynomials in $${\mathbb{F}_p}\left[ X \right]$$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of ℤK lying above p. We further specify for every prime ideal of ℤK lying above p, the ramification index, the residue degree, and a p-generator.

On power integral bases for certain pure number fields defined by

Let be a pure number field generated by a root α of a monic irreducible polynomial with is a square free integer, r and s are two positive integers. In this article, we study the monogenity of K. We prove that if and then K is monogenic. We give also sufficient conditions on r, s, and m for K to be not monogenic. Some illustrating examples are given too.

2-Orthogonal polynomials and Darboux transformations. Applications to the discrete Hahn-classical case

In this paper, we are interested in the following problem. We assume that is a monic 2-orthogonal polynomial sequence and we analyse the existence of a sequence of 2-orthogonal polynomials such that we have a decomposition This constitutes the counterpart in the framework of 2-orthogonality of those associated with the Christoffel transformation in the standard case. Some illustrative examples are shown.