多项式判别矩阵的若干性质及其应用

具有文字系数的多项式f(x)，其判别矩阵是f与f′的Sylvester矩阵通过添加一行一列而得，已经知道，判别矩阵的偶数阶主子式的符号确定了f(x)的相异根(实根、复根)的数目，这里介绍如何将奇数阶与偶数阶主子式相结合用以判定该多项式的相异负根或正根的数目，并进一步判定其在区间上的实根数，本文还研究了与判别矩阵相关的一些实用性质，并应用这些性质给出了4次键合多项式不能正分解的一组简洁的充分必要条件。     

一元复系数多项式实根的判定

对于一元实系数多项式的实根问题,运用斯图姆(Sturm)方法不仅可以确定其实根的个数以及正负根的个数,而且对于任意给定的区间(a,b)可以确定这个多项式在此区间内实根的个数,但是对于一元复系数多项式呢?本文给出一般方法把一元复系数多项式的实根问题归     

动力系统中心焦点的判定方法

给出了判定微分动力系统平衡点类型的一种方法.该方法通过Dulac函数将平衡点类型的判定转化为多项式方程组实根求解,后者可由符号计算来完成.通过结合Bendixson-Dulac定理、积分因子法、李雅普诺夫方法,给出最终判定结论.     

二次方程根的判别定理的可逆性

初中代数介绍了一元二次方程实根个数的判定定理: 一元二次方程ax~2+bx+c=0,称△=b~2-4ac为根的判别式,当△>0时,方程有两个不等的实根; △=0时,方程有两个相等的实根; △<0时,方程没有实数根。这个定理是个分断式命题,三个分支中的条件和结论是极为显见的,即由判别式的符号来判定实根的个数,然而教材中的习题却用到由实根的个数来确定判别式的符号。     

区间多项式的鲁棒稳定性

本文首先给出了一种新的判定多项式稳定的充要条件（引理2.1）.然后,在此基础上,研究了区间多项式的鲁棒稳定性,得到了若干判别区间多项式的充分条件（定理2.2-定理2.3）.由于所得的摄动界完全可由原末被扰动的多项式的系数所决定,这使得本文的方法比现有的结果简单好用.文末的例子说明了本文方法的有效性.     

多项式的一个性质及其在计算根时的应用

<正> 求多项式的根,方法很多.但大多数方法要从所求根的某个足够好的近似值出发,因而事先要进行根的隔离与初估.通常,对实系数多项式,可用施斗姆定理隔离实根,用黎斯定理隔离复根.当然,也可以不预先隔离,而用罗巴切夫斯基方法直接去求各根的近似值.但这些方法,其叙述、证明、计算程序都是相当复杂的.     

关于多项式与多项式函数相等的注记

给出了域上两个多项式作为多项式相等与作为多项式函数恒等的充要条件,引入有限域上约化多项式的概念,给出了有限域上多项式函数重根的定义及判定法则.     

多项式的重根判别式和代数基本定理的简证

运用齐次线性方程组的理论研究实数域上多项式根的问题,给出了n次实系数多项式在复数域上存在某种特殊非零重根的判别公式,同时给出了代数基本定理的一个简洁的代数证明.     

用二分法求解一元实系数多项式方程的全部实根

运用二分法,结合实系数多项式零点的界定理及Sturm定理,给出了一个求解一元实系数多项式方程全部实根的实用数值方法.     

关于$5$次对称形式正性的机器判定

利用参系数多项式正实根的判别序列,给出了多变元5次对称形式在Rn 上取非负值的显示判定方法.并以此为依据,导出了一个有效的算法,能够在变元数较多时也可以使用计算机来自动判定.     

Number of zeros of interval polynomials

In this paper, we develop a rigorous algorithm for counting the real interval zeros of polynomials with perturbed coefficients that lie within a given interval, without computing the roots of any polynomials. The result generalizes Sturm’s Theorem for counting the roots of univariate polynomials to univariate interval polynomials.     

Two finite hypergeometric sequences of discrete orthogonal polynomials

Two finite hypergeometric sequences of symmetric orthogonal polynomials of a discrete variable are introduced and their standard properties, such as second-order difference equations, explicit forms of the polynomials and three term recurrence relations are obtained. As a consequence of two specific Sturm–Liouville problems, it is proved that these polynomials are finitely orthogonal with respect to two symmetric weight functions.     

Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions

Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.     

Multivariate stable Eulerian polynomials on segmented permutations

Recently, Nunge studied Eulerian polynomials on segmented permutations, namely generalized Eulerian polynomials, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunge’s conjecture. Our proof is based on Brändén’s stable multivariate Eulerian polynomials. By acting on Brändén’s polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunge’s conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.     

A finite sequence of Hahn-type discrete orthogonal polynomials

ABSTRACT

By considering a specific Sturm–Liouville problem, we introduce a finite sequence of Hahn-type discrete polynomials and prove that they are finitely orthogonal on the real line. We then compute their norm square value by using Dougall's bilateral sum and obtain all moments corresponding to the introduced polynomials.     

Uniqueness of reconstruction of the Sturm–Liouville problem with spectral polynomials in nonseparated boundary conditions

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.     

Simple global minimization algorithm for one-variable rational functions

Sturm's chain technique for evaluation of a number of real roots of polynomials is applied to construct a simple algorithm for global optimization of polynomials or generally for rational functions of finite global minimal value. The method can be applied both to find the global minimum in an interval or without any constraints. It is shown how to use the method to minimize globally a truncated Fourier series. The results of numerical tests are presented and discussed. The cost of the method scales as the square of the degree of the polynomial.     

Single precision computation of the sign of algebraic predicates

We simplify and improve our techniques of the association of long integers with polynomials for computations in the ring of integers and apply these techniques to the computation of the signs of matrix determinants, Sturm sequences, and other algebraic and geometric predicates.