Poles of the topological zeta function for plane curves and Newton polyhedra

The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For functions that are nondegenerate with respect to their Newton polyhedron it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, W. Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this Note we show how to determine from the Newton polyhedron of a nondegenerate plane curve which candidate poles are actual poles. To cite this article: A. Lemahieu, L. Van Proeyen, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The barycentric weights of rational interpolation with prescribed poles

We introduce a method for calculating rational interpolants when some (but not necessarily all) of their poles are prescribed. The algorithm determines the weights in the barycentric representation of the rationals; it simply consists in multiplying each interpolated value by a certain number, computing the weights of a rational interpolant without poles, and finally multiplying the weights by those same numbers. The supplementary cost in comparison with interpolation without poles is about (v + 2)N, where v is the number of poles and N the number of interpolation points. We also give a condition under which the computed rational interpolation really shows the desired poles.     

Calculation of poles of meromorphic functions with q-d,r-s and ε-algorithms. Acceleration of these processes

We study (Sections 1 and 2) the speed of convergence of the q-d algorithm to poles of a meromorphic function in order to accelerate it.In Section 3 and 4, we show that the quotient of two successive vertical terms of two well known algorithms (r-s and ε) approaches also these poles. At last, a theorem of acceleration is given.     

Abschätzung des Restintegrals bei der Verhaltensfunktion eines Regelsystems

Summary The line integral obtained by the use of the Inversion Theorem of the Laplace Transformation is usually evaluated by applying the calculus of residues. This method can also be applied to find the behavior function of a regulation circuit from its Laplace Transform, if neither the regulator nor the process has a dead time. If there is a dead time in the regulation circuit, the Laplace Transform of the behavior function has an infinite number of poles. Therefore it is very difficult to evaluate exactly the inversion integral, and it is only possible if the arising infinite sum is convergent. In the present paper it is shown that in this case we nevertheless can apply the calculus of residues, because we thus obtain a good approximation applying it only to a finite number of poles. Thus the sum of the residues of a finite number of poles is proved to be a function approximating asymptoticly the exact function for large values of the time. But it must be assumed that always all the poles which are on the right side of a parallel to the imaginary axis are used.

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Dissertation, ausgeführt am Mathematischen Institut der Universität Zürich, unter Leitung von Herrn Prof. Dr.B. L. Van der Waerden.     

The Linear Rational Pseudospectral Method with Preassigned Poles

We present a linear rational pseudospectral (collocation) method with preassigned poles for solving boundary value problems. It consists in attaching poles to the trial polynomial so as to make it a rational interpolant. Its convergence is proved by transforming the problem into an associated boundary value problem. Numerical examples demonstrate that the rational pseudospectral method is often more efficient than the polynomial method.     

Convergence rates of a family of barycentric osculatory rational interpolation

It is well-known that osculatory rational interpolation sometimes gives better approximation than Hermite interpolation, especially for large sequences of points. However, it is difficult to solve the problem of convergence and control the occurrence of poles. In this paper, we propose and study a family of barycentric osculatory rational interpolation function, the proposed function and its derivative function both have no real poles and arbitrarily high approximation orders on any real interval.     

三元重心Thiele型混合有理插值

有理插值比多项式插值有更好的近似,但有理插值一般很难控制极点的产生.基于Thiele型连分式插值与重心有理插值,构造三元重心Thiele型混合有理插值,当选取适当的权后能避免部分极点的产生.文章最后通过数值例子验证了这种方法的正确性和有效性.     

一类极点的留数计算

本文探讨了一类比较特殊的极点类型的留数计算并从理论上给出证明.     

Barycentric rational interpolation with no poles and high rates of approximation

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of the distribution of the points. These interpolants depend linearly on the data and include a construction of Berrut as a special case.     

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

51.IntroductionLetF.beafinitefieldofqele~swithcharederisticp.LetXbeanThdilnensionalalgebraicsetdeadedoverF..ThezetafunctionofX/F.isdennedbywhereXOdenotesthesetofclosedpointsOfX/F.and#X(F.d)denotesthenUInberof.F.d-rationalpointsonX.ItiseasytoseethatZ(X,T)isapowerserieswithintegercoefficients.Dwork'srationalltytheoremIg]showsthatthezetafunctionZ(X,T)isrationalinT.ThuS,therearealgebraicintegerspll'.3Pr,pl,'3basuchthatThereisagoodreasonthatwemoantheabovezetafunctionbythepower(--1)"-…     

Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000     

The scattering of a plane wave by a trap in the critical case

The scattering of a plane wave by a resonator with a narrow coupling channel is considered. The velocity potential of the scattered wave in this resonator has two series of poles with small imaginary parts, corresponding to the main trap and the coupling channel, the effect of which inside the trap differs by an order of magnitude. The critical case, when the limiting value for the poles from both series is the same, is investigated. It is shown that in this case two poles exist, which converge to this limiting value, and they both inherit resonance properties, characteristic for poles generated by the main trap. The principal terms of the asymptotic forms of the poles and the scattered wave are constructed.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

Monodromization and Difference Equations with Meromorphic Periodic Coefficients

We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.     

Rational matrix and operator functions with prescribed singularities

In this paper it is shown how a rational matrix function may be reconstructed when complete information about its zeros and poles is given. The analogous problem for infinite dimensional operator functions is also solved.     

How poles of orthogonal rational functions affect their Christoffel functions

We show that even a relatively small number of poles of a sequence of orthogonal rational functions approaching the interval of orthogonality, can prevent their Christoffel functions from having the expected asymptotics. We also establish a sufficient condition on the rate for such asymptotics, provided the rate of approach of the poles is sufficiently slow. This provides a supplement to recent results of the authors where poles were assumed to stay away from the interval of orthogonality.     

Elliptic grids, rational functions, and the Padé interpolation

We consider rational functions with n prescribed poles for which there exists a divided difference operator transforming them to rational functions with n−1 poles. The poles of such functions are shown to lie on the elliptic grids. There is a one-to-one correspondence between this problem of admissible grids and the Poncelet problem on two quadrics. Additionally, we outline an explicit scheme of the Padé interpolation with prescribed poles and zeros on the elliptic grids. Dedicated to Richard Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—42C05; Secondary—39A13, 41A05, 41A21.     

一类收敛的线性Hermite重心权有理插值

众所周知, Hermite有理插值比Hermite多项式插值具有更好的逼近性, 特别是对于插值点序列较大时, 但很难解决收敛性问题和控制实极点的出现. 本文建立了一类线性Hermite重心有理插值函数$r(x)$,并证明其具有以下优良性质: 第一, 在实数范围内无极点; 第二, 当$k=0,1,2$时,无论插值节点如何分布, 函数$r^{(k)}(x)$具有$O(h^{3d+3-k})$的收敛速度; 第三, 插值函数$r(x)$仅仅线性依赖于插值数据.