On the Structure of the Set of Zeros of Quaternionic Polynomials

We prove that any quaternionic polynomial (with the coefficients on the same side) has two types of zeroes: the zeroes are either isolated or spherical ones, i.e., those ones which form a whole sphere. What is more, the total quantity of the isolated zeroes and of the double number of the spheres does not outnumber the degree of the polynomial.     

ON SETS OF ZEROES OF CLIFFORD ALGEBRA-VALUED POLYNOMIALS

In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjugate ones. The total number of zeroes does not exceed the degree of the polynomial. We also present a technique for computing the zeroes.     

The fundamental theorem of algebra for Hamilton and Cayley numbers

In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological argument due to Gauss. G. Gentili and F. Vlacci are partially supported by G.N.S.A.G.A. of the I.N.D.A.M. and by M.I.U.R.     

On Reduction of Quaternionic Polynomials and Their Identical Equality

We show that any quaternionic polynomial with one variable can be represented in such a way that the number of its terms will be not larger than a certain number depending on the degree of the polynomial. We study also some particular cases where this number can be made even smaller. Then we use the above-mentioned representation to study how to check whether two given quaternionic polynomials with one variable are identically equal. We solve this problem for all linear polynomials and for some types of nonlinear polynomials.     

Optimal comparison of the <Emphasis Type="Italic">p</Emphasis>-norms of Dirichlet polynomials

In this sequel to Part I of this series [8], we present a different approach to bounding the expected number of real zeroes of random polynomials with real independent identically distributed coefficients or more generally, exchangeable coefficients. We show that the mean number of real zeroes does not grow faster than the logarithm of the degree. The main ingredients of our approach are Descartes’ rule of signs and a new anti-concentration inequality for the symmetric group. This paper can be read independently of part I in this series.     

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

We study in detail the zero set of a slice regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.     

A representation formula of sum rules for zeros of polynomials

A representation formula in terms of Lucas polynomials of the second kind in several variables (see formula (4.3)), for the sum rulesJ _s ⁽ⁱ⁾ introduced by K.M. Case [1] and studied by J.S. Dehesa et al. [2]–[3] in order to obtain informations about the zeros’ distribution of eigenfunctions of a class of ordinary polynomial differential operator, is derived. Lavoro eseguito nell’ambito del G.N.I.M. del C.N.R.     

Generating approximate parametric roots of parametric polynomials

Built upon a ground field is the parametric field, the Puiseux field, of semi-terminating formal fractional power series. A parametric polynomial is a polynomial with coefficients in the parametric field, and roots of parametric polynomials are parametric. For a parametric polynomial with nonterminating parametric coefficients and a target accuracy, using sensitivity of the Newton Polygon process, a complete set of approximate parametric roots, each meeting target accuracy, is generated. All arguments are algebraic, from the inside out, self-contained, penetrating, and uniform in that only the Newton Polygon process is used, for both preprocessing and intraprocessing. A complexity analysis over ground field operations is developed; setting aside root generation for ground field polynomials, but bounding such, polynomial bounds are established in the degree of the parametric polynomial and the target accuracy.     

The Weierstrass factorization theorem for slice regular functions over the quaternions

The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains natural quaternionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular functions. This result holds in a formulation that reflects the peculiarities of the quaternionic setting and the structure of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions.     

An alternative class of irreducible polynomials for optimal extension fields

Optimal extension fields (OEF) are a class of finite fields used to achieve efficient field arithmetic, especially required by elliptic curve cryptosystems (ECC). In software environment, OEFs are preferable to other methods in performance and memory requirement. However, the irreducible binomials required by OEFs are quite rare. Sometimes irreducible trinomials are alternative choices when irreducible binomials do not exist. Unfortunately, trinomials require more operations for field multiplication and thereby affect the efficiency of OEF. To solve this problem, we propose a new type of irreducible polynomials that are more abundant and still efficient for field multiplication. The proposed polynomial takes the advantage of polynomial residue arithmetic to achieve high performance for field multiplication which costs O(m ^3/2) operations in \mathbbF_p{\mathbb{F}_p} . Extensive simulation results demonstrate that the proposed polynomials roughly outperform irreducible binomials by 20% in some finite fields of medium prime characteristic. So this work presents an interesting alternative for OEFs.     

多项式组的特征分解

任一多项式理想的特征对是指由该理想的约化字典序Grobner基G和含于其中的极小三角列C构成的有序对(G,C).当C为正则列或正规列时,分别称特征对(G,C)为正则的或正规的.当G生成的理想与C的饱和理想相同时,称特征对(G,C)为强的.一组多项式的(强)正则或(强)正规特征分解是指将该多项式组分解为有限多个(强)正则或(强)正规特征对,使其满足特定的零点与理想关系.本文简要回顾各种三角分解及相应零点与理想分解的理论和方法,然后重点介绍(强)正则与(强)正规特征对和特征分解的性质,说明三角列、Ritt特征列和字典序Grobner基之间的内在关联,建立特征对的正则化定理以及正则、正规特征对的强化方法,进而给出两种基于字典序Grobner基计算、按伪整除关系分裂和构建、商除可除理想等策略的(强)正规与(强)正则特征分解算法.这两种算法计算所得的强正规与强正则特征对和特征分解都具有良好的性质,且能为输入多元多项式组的零点提供两种不同的表示.本文还给出示例和部分实验结果,用以说明特征分解方法及其实用性和有效性.     

Cubic formulas for computing the zeros of certain quaternionic polynomial

In this paper, we derive the explicit formulas for computing the zeros of certain cubic quaternionic polynomial. From these, we obtain a necessary and sufficient condition to quaternionic cubic polynomial have a spherical zero, and some examples are also provided. Moreover, we will discuss some applications of the cubic quaternionic formulas. Copyright © 2017 John Wiley & Sons, Ltd.     

Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.     

On approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces

This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f(x) ∈ LM*[0,1], changes its sign at most once in (0,1), then there exists x0 ∈ (0,1) and a polynomial Pn ∈ Πn(+) such that f (x) -Pn (x)x-x0 M ≤ Cω( f,n-1/2 )M, where Πn(+) indicates the set of all polynomials of degree n with positive coefficients.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

Analysis of least squares finite element methods for a parameter-dependent first-order system *

We consider polynomials orthagonal with respect to a measure μ with an absolutely continuous component and a finite discrete part. We prove that subject to certatin integrability conditions, the polynomials satisfy a second order differential equation. The zeroes of such polynomials determine the equilibrium position of movable n unit charges in an external field determined by the measure μ. We also evaluate the discriminant of such orthagonal polynomials and use it to compute the total energy of the system at equilibrium in terms of the recursion coefficients of the orthonormal polynomials. We also investigate several explicit models, the Koornwinder polynomials, the Ginzburg-Landau potential and the generalized Jacobi weights.     

Oscillation of analytic curves

The number of zeroes of the restriction of a given polynomial to the trajectory of a polynomial vector field in , in a neighborhood of the origin, is bounded in terms of the degrees of the polynomials involved. In fact, we bound the number of zeroes, in a neighborhood of the origin, of the restriction to the given analytic curve in of an analytic function, linearly depending on parameters, through the stabilization time of the sequence of zero subspaces of Taylor coefficients of the composed series (which are linear forms in the parameters). Then a recent result of Gabrielov on multiplicities of the restrictions of polynomials to the trajectories of polynomial vector fields is used to bound the above stabilization moment.

     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

Zolotarev problem in the metric of L1([−1, 1])

In this paper we solve the problem of the determination of a polynomial of degree n with given two leading coefficients which has the least deviation from zero in the metric of L₁ ([?1, 1]). The extremal polynomial is expressed in the form of some linear combination of Chebyshev polynomials of the second kind.