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.     

Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball

We find a polynomial solution of the Dirichlet problem for the inhomogeneous biharmonic equation with polynomial right-hand side and polynomial boundary data in the unit ball. To this end, we use the closed-form representation of harmonic functions in the Almansi formula.     

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.     

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.     

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.     

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].     

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.     

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.     

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 numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

Characterization of the convergence domains of polynomial series and the minimal convergence domain

A characterization of the convergence domains of polynomial series is disucssed, the minimal convergence domain for a kind of polynomial series is shown.     

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.     

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.     

A single oval of Cassini for the zeros of a polynomial

We derive two ovals of Cassini, each containing all the zeros of a polynomial. The computational cost to obtain these ovals is similar to that of the Brauer set for the companion matrix of a polynomial, although they are frequently smaller. Their derivation is based on the Gershgorin set for an appropriate polynomial of the companion matrix.     

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.     

Constructions of general polynomial lattice rules based on the weighted star discrepancy

In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration.Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy.In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy.     

探讨Abel积分零点的一个定理

本文研究了具有中心环域的可积多项式系统,探讨此系统在用多项式进行微扰的情况下,其所对应的Abel积分的零点个数的计算方法,给出了一个较为实用的定理,并举出了若干个应用的例子．     

Stability of the Cauchy problem for a multidimensional difference operator and the amoeba of the characteristic set

Employing the notion of the amoeba of an algebraic set we state a multidimensional analog of the following condition: The moduli of roots of a polynomial are less than one. This analog is proved to be a necessary and sufficient condition for stability of the Cauchy problem for a polynomial difference operator with constant coefficients.