Algebraic Polynomials Research Articles

On the behaviour of derivative of algebraic polynomials in the regions with cusps

ABSTRACT In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region.

Classification of noncommutative conics associated to symmetric regular superpotentials

Let [Formula: see text] be a [Formula: see text]-dimensional quantum polynomial algebra, and [Formula: see text] a central regular element. The quotient algebra [Formula: see text] is called a noncommutative conic. For a noncommutative conic [Formula: see text], there is a finite-dimensional algebra [Formula: see text] which determines the singularity of [Formula: see text]. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, [Formula: see text] is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs [Formula: see text], and calculate the algebras [Formula: see text].

Factorization of polynomials with rational exponents over a field

The purpose of this paper is to study some factorization properties in the polynomial algebra with rational exponents, over a field K. Among other things, we investigate some criteria for irreducibility and the idf-property in As an application, we get a factorization theorem in

Higher-order Image Conjecture for differential operators of polynomial algebras

Higher-order Image Conjecture for differential operators of polynomial algebras

Infinite symmetric products of rational algebras and spaces

We show that the infinite symmetric product of a connected graded-commutative algebra over ℚ is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over ℚ is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold–Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded-commutative algebras are free; this allows us to determine minimal Sullivan models for finite symmetric products of complex projective spaces.

Projective Limit Techniques for the Infinite Dimensional Moment Problem

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra A be represented as an integral with respect to a Radon measure on the character space X(A) of A equipped with the Borel sigma -algebra generated by the weak topology? We approach this problem by constructing X(A) as a projective limit of the character spaces of all finitely generated unital subalgebras of A. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on A with respect to a measure on the cylinder sigma -algebra on X(A) (resp. a Radon measure on the Borel sigma -algebra on X(A)) provided that for any finitely generated unital subalgebra of A the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on A. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a “partially Archimedean” quadratic module of A. Our results in particular apply to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem.

A gyroscopic polynomial basis in the sphere

Standard spectral codes for full sphere dynamics utilize a combination of spherical harmonics and a suitable radial basis to represent fluid variables. These basis functions have a rotational invariance not present in geophysical flows. Gyroscopic alignment - alignment of dynamics along the axis of rotation - is a hallmark of geophysical fluids in the rapidly rotating regime. The Taylor-Proudman theorem, resulting from a dominant balance of the Coriolis force and the pressure gradient force, yields nearly invariant flows along this axial direction. In this paper we tailor a coordinate system to the cylindrical structures found in rotating spherical flows. This “spherindrical” coordinate system yields a natural hierarchy of basis functions, composed of Jacobi polynomials in the radial and vertical direction, regular throughout the ball. We expand fluid variables using this basis and utilize sparse Jacobi polynomial algebra to implement all operators relevant for partial differential equations in the spherical setting. We demonstrate the representation power of the basis in three eigenvalue problems for rotating fluids.

Explicit Generators of the Centre of the Quantum Group

A finite generating set of the centre of any quantum group is obtained, where the generators are given by an explicit formulae. For the slightly generalised version of the quantum group which we work with, we show that this set of generators is algebraically independent, thus the centre is isomorphic to a polynomial algebra.

The non-commutative Korteweg–de Vries hierarchy and combinatorial Pöppe algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show that within the class of polynomial partial differential fields, at every order, each member of the non-commutative potential Korteweg–de Vries hierarchy is unique. Indeed, we prove to all orders, that each such member matches the non-commutative Lax hierarchy field, which is therefore a polynomial partial differential field. We achieve this by constructing the abstract combinatorial algebra that underlies the non-commutative potential Korteweg–de Vries hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by the set of all compositions endowed with the Pöppe product. This product is the abstract representation of the product rule for Hankel operators pioneered by Ch. Pöppe for integrable equations such as the Sine–Gordon and Korteweg–de Vries equations. Integrability of the hierarchy members translates, in the combinatorial algebra, to proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’. Proving the existence of such Pöppe polynomial expansions boils down to solving a linear algebraic problem for the expansion coefficients, which we solve constructively to all orders.

Twisted conjugacy in $${\mathrm {GL}}_2$$ and $${\mathrm {SL}}_2$$ over polynomial algebras over finite fields

Twisted conjugacy in $${\mathrm {GL}}_2$$ and $${\mathrm {SL}}_2$$ over polynomial algebras over finite fields

-graded identities of the Lie algebras in characteristic 2

AbstractLetKbe any field of characteristic two and let$U_1$and$W_1$be the Lie algebras of the derivations of the algebra of Laurent polynomials$K[t,t^{-1}]$and of the polynomial ringK[t], respectively. The algebras$U_1$and$W_1$are equipped with natural$\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of$U_1$and$W_1$, and we prove that they do not admit any finite basis.

One-Sided Multiplier Approximation of Unbounded Functions

The main objective of this article is to study the degree of best one-sided multiplier approximation of unbounded functions ,( ),[ 1,1]npgLY Y= −by means of the average modulus of smoothness by using sequences of algebraic polynomials Pn of degree less than n, nr+ 1, also in this search we shall prove a direct theorem by sequences Pn and some results.

On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras

In this paper, we study the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras.

On Chemical Invariants of Semitotal-Point Graph and Its Line Structure of the Acyclic Kragujevac Network: A Novel Mathematical Analysis

Acyclic Kragujevac network is denoted by K ; K ∈ K g q = r 2 s + 1 + 1 , r . In this article, we have taken a deep look at some of the topological properties of the semitotal-point graph as well as its line structure by computing some algebraic polynomials. Few degree-dependent chemical invariants that can primarily be obtained via these algebraic expressions are also put forward.

Best multi Approximation of Unbounded Functions by Using Modulus of Smoothness

We present an estimate of the degree of best multi approximation of unbounded function on 〖[-1,1]〗^d by algebraic polynomials in weighted space. The studied of the relation between the best approximation of derivatives functions in weighted space and the best approximation of unbounded functions in the same space.

Generalized down-up algebras revisited from a viewpoint of Gröbner basis theory

The so called generalized down-up algebras are revisited from a viewpoint of Gröbner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided ), and by means of homogeneous Gröbner defining relations, the associated graded structures of generalized down-up algebras, namely the associated graded algebras, Rees algebras, and the homogenized algebras of generalized down-up algebras, are explored comprehensively.

Algebraic Basis of the Algebra of All Symmetric Continuous Polynomials on the Cartesian Product of ℓp-Spaces

We construct a countable algebraic basis of the algebra of all symmetric continuous polynomials on the Cartesian product ℓp1×…×ℓpn, where p1,…,pn∈[1,+∞), and ℓp is the complex Banach space of all p-power summable sequences of complex numbers for p∈[1,+∞).

Stable high-order randomized cubature formulae in arbitrary dimension

We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure μ defined on a domain Γ⊆Rd, in any dimension d. Each cubature formula is exact on a given finite-dimensional subspace Vn⊂L2(Γ,μ) of dimension n, and uses pointwise evaluations of the integrand function ϕ:Γ→R at m>n independent random points. These points are drawn from a suitable auxiliary probability measure that depends on Vn. We show that, up to a logarithmic factor, a linear proportionality between m and n with dimension-independent constant ensures stability of the cubature formula with high probability. We also prove error estimates in probability and in expectation for any n≥1 and m>n, thus covering both preasymptotic and asymptotic regimes. Our analysis shows that the expected cubature error decays as n/m times the L2(Γ,μ)-best approximation error of ϕ in Vn. On the one hand, for fixed n and m→∞ our cubature formula can be seen as a variance reduction technique for a Monte Carlo estimator, and can lead to enormous variance reduction for smooth integrand functions and subspaces Vn with spectral approximation properties. On the other hand, when n,m→∞, our cubature becomes of high order with spectral convergence. As a further contribution, we analyse also another cubature whose expected error decays as 1/m times the L2(Γ,μ)-best approximation error of ϕ in Vn, but with constants that can be larger in the preasymptotic regime. Finally we show that, under a more demanding (at least quadratic) proportionality between m and n, all the weights of the cubature are strictly positive with high probability. As an example of application, we discuss the case where the domain Γ has the structure of Cartesian product, μ is a product measure on Γ and Vn contains algebraic multivariate polynomials.

Algebra of polynomials bounded on some strips

In the present paper, for a finite sequence of single variable polynomials g (x) = (g1(x), g2(x), ..., gn(x)), we study the ring of geometrically bounded elements on a generalized strip Mc(g) in ℝn+1 which is the solution of the system of polynomial inequalities g1(x) £ y1 £ g1(x) + c1, g2(x) £ y2 £ g2(x) + c2 …, gn(x) £ yn £ gn(x) + cn. This ring is shown to be the finitely generated ℝ-algebra ℝ[y1 - g1(x), y2 - g2(x), ..., yn - gn(x)] provided that c = (c1, c2, ..., cn) is a positive vector. However, if c = 0 then this algebra is not finitely generated in general. In particular, we point out that the ring of geometrically bounded elements on ‘a generalized strip’ of the form M(g1, g2) in ℝ2 which is the solution of the polynomial inequality g1(x) £ y £ g2(x) is trivial (i.e., is equal to ℝ) provided that g1(x) is less than g2(x) at infinity. As a consequence, we can describe the ring of geometrically bounded elements on a finite union of disjoint strips

The q-Onsager algebra and its alternating central extension

The q-Onsager algebra Oq has a presentation involving two generators W0, W1 and two relations, called the q-Dolan/Grady relations. The alternating central extension Oq has a presentation involving the alternating generators {W−k}k=0∞, {Wk+1}k=0∞, {Gk+1}k=0∞, {G˜k+1}k=0∞ and a large number of relations. Let 〈W0,W1〉 denote the subalgebra of Oq generated by W0, W1. It is known that there exists an algebra isomorphism Oq→〈W0,W1〉 that sends W0↦W0 and W1↦W1. It is known that the center Z of Oq is isomorphic to a polynomial algebra in countably many variables. It is known that the multiplication map 〈W0,W1〉⊗Z→Oq, w⊗z↦wz is an isomorphism of algebras. We call this isomorphism the standard tensor product factorization of Oq. In the study of Oq there are two natural points of view: we can start with the alternating generators, or we can start with the standard tensor product factorization. It is not obvious how these two points of view are related. The goal of the paper is to describe this relationship. We give seven main results; the principal one is an attractive factorization of the generating function for some algebraically independent elements that generate Z.