On the size of multivariate polynomial lemniscates and the convergence of rational approximants

In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest.     

Integral estimates of lengths of level lines of rational functions and zolotarev’s problem

Integral estimates of lengths of level lines (lemniscates) of rational functions of a complex variable are obtained. These estimates are related to the problem of separation of compact sets by rational functions and to Zolotarev’s problem.     

The logarithmic energy of zeros and poles of a rational function

On assuming that certain lemniscates of a rational function are connected, we establish some sharp inequality that involves the logarithmic energy of a discrete charge concentrated at the zeros and poles of this function and the absolute values of its derivatives at these points. The equality in this estimate is attained for specially arranged zeros and poles of a suitable Zolotarev fraction and for special distributions of charge.     

An extremal problem for the derivative of a rational function

Erd?s’ well-known problem on the maximum absolute value of the derivative of a polynomial on a connected lemniscate is extended to the case of a rational function. Moreover, under the assumption that certain lemniscates are connected, a sharp upper bound for the absolute value of the derivative of a rational function at any point in the plane different from the poles is found. The role of the extremal function is played by an appropriate Zolotarev fraction.     

A property of the planar measure of the lemniscates

In this paper, we establish the following conjecture: There exists a constant K such that every lemniscate E(α,c), α∈Cn, c>0, contains a disk B(α,c) with μ(E(α,c))?Kμ(B(α,c)), where μ is the planar measure. We prove this conjecture for any family of lemniscates with at the most three foci and for any family of lemniscates where its foci satisfy a suitable condition.     

Asymptotic Bernstein inequality on lemniscates

In this paper we examine the Bernstein-Markov inequality on special compact subsets of the complex plane, namely on lemniscates. Sharp constants are obtained which involve the Green function of the complement and the density of equilibrium measure of the compact set. Using lemniscates is a useful tool because of the possibility of taking inverse images. The proof also uses so-called peaking polynomials which will be constructed.     

Splice diagram determining singularity links and universal abelian covers

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their universal abelian covers are homeomorphic. To prove the last theorem we have to generalize our notions to orbifolds.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

关于有理模和余理想子代数的性质

In this paper, for some used conceptions and notations, we see ［1］ and ［2］.§1. Rational Module and Its Exact Sequence In ［1］, Cai Chuanreng and Cheng Huixiang have proved that relative Hopf modules and rational modules are one by one corresponding. In ［2］, Zhang Liangyun has given the dual relationship between relative Hopf modules. Naturally, we have a question to ask: is the dual module of a rational module still a rational module? This answer is affirmative. Let H be a Hopf …     

Hyperellipticity of offsets to rational plane curves

In this paper, we prove irreducible offsets to rational plane curves are hyperelliptic in general and compute the genus of them. We also give a criterion for deciding the irreducibility of offsets to rational plane curves.     

Application du prolongement analytique au problème inverse du potentiel

Sommaire Le but de cet article est établir quelques résultats nouveaux sur le problème inverse du potentiel newtonien. Nous démontrons deux théorèmes d'unicité: pour les polyédres convexes dansR ⁿ et pour les lemniscates dansR ². L'instrument principal est un lemme basé sur une idée de V. Kondrachkov rarement utilisé malgré sa puissance. Nous montrons son efficacité en liaison avec la méthode du prolongement analytique des potentiels.

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The goal of this paper is to establish some new results in the inverse Newtonian potential problem. We prove two uniqueness theorems: for convex polyhedra inR ⁿ and for lemniscates inR ². The main tool is a lemma based upon an idea of V. Kondrashkov which, though powerful, is rarely used. We show its efficiency applied together with the method of analytic continuation of potentials.

     

Rational energy decay rate for a wave equation with dynamical control

We consider a wave equation with dynamical control. We first establish the rational energy decay rate using a multiplier method. Next, using a spectrum method, we prove that the rational energy decay rate is optimal.     

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

Hyperbolic dimension and Poincaré critical exponent of rational maps

We study the Poincaré series of rational maps. By investigating the property of conical Julia set and dissipative measure, we prove that the Poincaré critical exponents are equal to the hyperbolic dimensions for a large class of rational maps.     

Ergodicity of invariant capacities

In this paper, we investigate capacity preserving transformations and their ergodicity. We obtain some limit properties under capacity spaces and then give the concept of ergodicity for a capacity preserving transformation. Based on this definition, we give several characterizations of ergodicity. In particular, we obtain a type of Birkhoff’s ergodic theorem and prove that the ergodicity of a transformation with respect to an upper probability is equivalent to a type of strong law of large numbers.     

Lemniscates 3D: a CAGD primitive?

A 3D lemniscate is an implicitly given surface which generalizes the well-known Bernoulli lemniscates curves and the Cassini ovals in 2D. It is characterized by placing a finite number of points in space (the foci) and choosing a constant (radius), its algebraic degree is twice the number of foci and it is always contained in the union of certain spheres centered at the foci. The distribution of the foci gives a rough idea of the 3D shapes that could be modeled with any of the connected components of the lemniscate. The position of the foci can be used to stretch and to produce knoblike features. Given a set of foci, for a small radius the lemniscate consists of a number of spherelike surfaces centered at the foci which do not touch each other. As the radius increases the disconnected pieces coalesce producing interesting surfaces. In order to make 3D lemniscates a potentially useful primitive for CAGD it is necessary to control the coalescing/splitting of the connected components of the lemniscate while we move the foci and change the radius, simultaneously. In this paper we offer tools towards this control. We look closely at the case of four noncoplanar foci. AMS subject classification 65D05, 65D17, 65D18This work was partially supported by grant G97 000651 of Fonacit, Venezuela.     

Continuous functions on the plane regular after one blowing-up

We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these functions correspond to rational functions which become regular after some blowings-up, we work on the plane where it suffices to blow-up points and then we can count the number of stages of blowings-up necessary. In the latest parts of the paper, we investigate the ring of rational continuous functions on the plane regular after one stage of blowings-up. In particular, we prove a Positivstellensatz without denominator in this ring.     

Small polynomials with integer coefficients

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of the integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for the integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for the integer Chebyshev constant. Moreover, all the bounds mentioned can be found numerically by using various extremal point techniques, such as the weighted Leja points algorithm. Applying our results in the classical case of the segment [0, 1], we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials. Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.     

Negative Dense Linear Orders

Considering dense linear orders, we establish their negative representability over every infinite negative equivalence, as well as uniformly computable separability by computable gaps and the productivity of the set of computable sections of their negative representations. We construct an infinite decreasing chain of negative representability degrees of linear orders and prove the computability of locally computable enumerations of the field of rational numbers.