Eisenstein判别法一种新的推广及应用

着力推广Eisenstein判别法,得到有关整系数多项式不可约的几个新的判别法,并应用这些新的判别法有效地判定一些不能用Eisenstein判别法判定的有理数域上的不可约多项式,及其有理根的存在性.     

对“Eisenstein判别法的应用（2）”一文中所承认的一个结论的商榷

在整系数不可约多项式中,有一类不可约多项式f_1(x),它们不能直接应用Eisenstein判别法来判别;一般教科书中都指出,这时可适当选取整数α、β,令x=αy+β,使g(y)=f_1(αy+β)能用Eisenstein判别法来判别。也有另一类不可约多项式(用S来表示这一类多项式)     

球面带形平移网络逼近的Jackson定理

研究了球面带型平移网络逼近阶用球面调和多项式的最佳逼近及光滑模的刻画问题．借助于球调和多项式的最佳逼近多项式和Riesz平均构造出了单位球面Sq上的带形平移网络,并建立了球面带形平移网络对Lp(Sq)中函数一致逼近的Jackson型定理．所得结果表明球面带形平移网络可以达到球调和多项式的逼近阶．     

整多项式可约性的一个判别法

整多项式可约性的一个判别法王琳（中央财院数学教研室）整系数多项式可约性的判定是多项式研究的一个基本问题，也是一个比较困难的问题．在这方面有著名的艾森斯坦因判别法．为论述方便先引人下面记号．设ｆ（ｘ）＝…＋ａ１ｘ＋ａｏ（ａｎ０）为一整系数多项式ｐ是一个...     

整系数多项式在整数环上不可约性的探讨

本文在艾森斯坦因判别法的基础上，对整系数多项式的次高项系数进行了讨论，得到了整系数多项式在整数环上不可约的一个新的判别法。     

整系数多项式在整数环上不可约性的进一步探讨

本文首先给出了整系数多项式有二次整系数多项式因式的一个必要条件，进而通过对整系数多项式ｆ（ｘ）＝ＡｎＸ２十αｎ－１Ｘｎ－１＋…＋αｏ中ｘｎ－２的系数αｎ－２的讨论，得到一类整系数多项式在整数环上是否可约的一个判别法。     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

整数环上的不可约多项式

著名的Eisenstein判别法为寻求整系数不可约多项式提供了方法,但此判别法的三个充分条件具有一定的局限性,致使对相当多的特殊整系数不可约多项式的判断失效.在总结前人研究工作的基础上,推导能有效判断特殊不可约整系数多项式的方法,拓展原有研究结果,可拓宽判断不可约整系数多项式的工具和方法.     

一类Turan型多项式序列的构造及其性质

应用实系数多项式的性质构造了一类满足Turan型不等式的多项式序列,证明了该多项式序列的几个性质,并给出了一些应用.     

多项式不可约性的一个定理

本文给出了判别有理数域上多项式不可约性的一个定理     

Difference Equations and Discriminants for Discrete Orthogonal Polynomials

We prove that any set of polynomials orthogonal with respect to a discrete measure supported on equidistant points contained in a half line satisfy a second order difference equation. We also give a discrete analogue of the discriminant and give a general formula for the discrete discriminant of a discrete orthogonal polynomial. As an application we give explicit evaluations of the discrete discriminants of the Meixner and the Hahn polynomials. A difference analogue of the Bethe Ansatz equations is also mentioned.Research partially supported by NSF grant DMS 99-70865     

On a class of hyperbolic polynomials

In the present paper, we investigate whether the roots of a biquadratic equation determined by a pair of real symmetric positive definite matrices of order 3 and a three-dimensional vector of parameters are real. We obtain the explicit representation of the discriminant of such a polynomial as the sum of at most two squares.     

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.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

Implementation of Pellet’s theorem

Pellet’s theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.     

d-Orthogonality of Humbert and Jacobi type polynomials

In this paper, we treat three questions related to the d-orthogonality of the Humbert polynomials. The first one consists to determinate the explicit expression of the d-dimensional functional vector for which the d-orthogonality holds. The second one is the investigation of the components of Humbert polynomial sequence. That allows us to introduce, as far as we know, new d-orthogonal polynomials generalizing the classical Jacobi ones. The third one consists to solve a characterization problem related to a generalized hypergeometric representation of the Humbert polynomials.     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.     

Equimodular curves

This paper is motivated by a problem that arises in the study of partition functions of Potts models, including as a special case chromatic polynomials. When the underlying graphs have the form of ‘bracelets’, the chromatic polynomials can be expressed in terms of the eigenvalues of a matrix. In this situation a theorem of Beraha, Kahane and Weiss asserts that the zeros of the polynomials approach the curves on which the matrix has two eigenvalues with equal modulus. It is shown here that (in general) these ‘equimodular’ curves comprise a number of segments, the end-points of which are the roots (possibly coincident) of a polynomial equation. The equation represents the vanishing of a discriminant, and the segments are in bijective correspondence with the double roots of another polynomial equation, which is significantly simpler than the discriminant equation. Singularities of the segments can occur, corresponding to the vanishing of a Jacobian. In addition, it is proved by algebraic means that the equimodular curves for a reducible matrix are closed curves. The question of dominance is investigated, and a method of constructing the dominant equimodular curves for a reducible matrix is suggested. These results are illustrated by explicit calculations in a specific case.     

Mixed discriminants

The mixed discriminant of $n$ Laurent polynomials in $n$ variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an $A$ -discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plücker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.     

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.