On the refinement of Cauchy's theorem and Pellet's theorem

Using the relationship of a polynomial and its associated polynomial, we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial. By employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function, we refine two classical results of Cauchy and Pellet about bounds of modules of polynomial zeros. Sufficient conditions are obtained for the polynomial whose Cauchy's bound and Pellet's bounds are strict bounds. The characteristics is given for the polynomial whose Cauchy's bound or Pellet's bounds can be achieved by the modules of zeros of the polynomial.     

A relation between the LG polynomial and the Kauffman polynomial

We give an explicit formula for the fact given by Links and Gould that a one variable reduction of the LG polynomial coincides with a one variable reduction of the Kauffman polynomial. This implies that the crossing number of an adequate link may be obtained from the LG polynomial by using a result of Thistlethwaite. We also give some evaluations of the LG polynomial.     

A note on the matching numbers of triangle-free graphs

For the sake of brevity the absolute values of the coefficients of the matching polynomial of a graph are called matching numbers in this note. It is shown that for a triangle-free graph these numbers coincide with the coefficients of the chromatic polynomial of its complement when this polynomial is written in factorial form. As an application it is mentioned that the coefficients of every rook polynomial are at the same time coefficients of some chromatic polynomial.     

Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial boundary data in the unit ball.     

关于对偶Steiner多项式的根的注记

受凸体的Steiner多项式的启发,定义了星体的对偶Steiner多项式,并利用对偶Aleksandrov-Fenchel不等式讨论了对偶Steiner多项式的根.进而,得到了关于对偶Steiner多项式的根的一些不等式,这些不等式恰好是关于Steiner多项式的根的不等式的对偶形式.     

Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems

We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.     

A note on the complexity of a phaseless polynomial interpolation

In this paper we revisit the classical problem of polynomial interpolation, with a slight twist; namely, polynomial evaluations are available up to a group action of the unit circle on the complex plane. It turns out that this new setting allows for a phaseless recovery of a polynomial in a polynomial time.     

Polynomial alternatives for the group of affine motions

It is known that every polycyclic-by-finite group – even if it admits no affine structure – allows a polynomial structure of bounded degree. A major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of , in contrast to the group of affine motions, is no longer a finite dimensional Lie group. In this paper we construct a family of (finite dimensional) Lie groups, even linear algebraic groups, of polynomial diffeomorphisms, which we call weighted groups of polynomial diffeomorphisms. It turns out that every polycyclic-by-finite group admits a polynomial structure via these weighted groups; in the nilpotent (and other) case(s), we can sharpen, by specifying a nice set of weights, the existence results obtained in earlier work. We introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding Mal`cev completion. This, and other properties, provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e.g. in the work of Fried, Goldman and Hirsch. Received November 30, 1998; in final form March 10, 1999     

Accurate Enclosure of the Zero Set of Multivariate Polynomials

The zero set of one general multivariate polynomial is enclosed by unions and intersections of funnel-shaped unbounded sets. There are sharper enclosures for the zero set of a polynomial in two complex variables with complex interval coefficients. Common zeros of a polynomial system can be located by an appropriate intersection of these enclosure sets in an appropriate space. The resulting domain is directly brought into polynomial equation solvers.     

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].     

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。     

On the reconstruction of graph invariants

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U-polynomial, the universal edge elimination polynomial ξ and the colored versions of the latter two are reconstructible.We also present a method of reconstructing boolean graph invariants, or in other words, proving recognizability of graph properties (of colored or uncolored graphs), using first order logic.     

Average case tractability of non-homogeneous tensor product problems with the absolute error criterion

We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.     

On the new fourth-order methods for the simultaneous approximation of polynomial zeros

A new iterative method of the fourth-order for the simultaneous determination of polynomial zeros is proposed. This method is based on a suitable zero-relation derived from the fourth-order method for a single zero belonging to the Schröder basic sequence. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is studied in detail for the proposed method. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. The construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed. Finally, numerical examples and the comparison with existing fourth-order methods are given.     

On the Pal‘s Problem

The correct answer of Pal‘s interpolation polynomial problem has been given is this paper,espe-cially,the explict form of this polynomial has been obtained.     

Algebraic computation of the number of zeros of a complex polynomial in the open unit disk by a polynomial representation

We present a general method for the exact computation of the number of zeros of a complex polynomial inside the unit disk, assuming that the polynomial does not vanish on the unit circle. We prove the existence of a polynomial sequence. This sequence involves a reduced number of arithmetic operations and the growth of intermediate coefficients remains controlled. We study the singular case where the constant term of a polynomial of this sequence vanishes.     

图和色多项式根2的阶

本文证明了3-连通非偶图的色多项式根2的阶为1;满足一定条件的非3-连通非偶图的色多项式根2的阶是图的非偶块和非偶可分块数.从而,把色多项式P(G)中1的阶是图G的非平凡块数这一结果进一步加以推广.     

Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.     

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ?rhus invariant of knots. Received February 14, 2000 / Published online October 11, 2000     

矩阵特征多项式的图论计算公式

给出了赋权有向图邻接矩阵特征多项式的图论计算公式,从而得到了一般矩阵特征多项式的图论计算方法,并且研究了赋权有向图邻接矩阵特征多项式和谱半径的一些性质．