Area and the inradius of lemniscates

We study geometric properties of filled lemniscates of a complex polynomial p(z) of degree n. In particular, we answer a question raised by H.H. Cuenya and F.E. Levis (2007) by showing that there is a constant C(n) such that for every lemniscate E(p,c). Here μ(E(p,c)) and r(E(p,c)) denote the area and the inradius of E(p,c).     

Length functions of lemniscates

We study metric and analytic properties of generalized lemniscates E _t (f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E _t (f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E _t (f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulae of the length function in some special cases.The author was supported the Göran Gustafsson foundation and grant RFBR no. 03-01-00304.The author was supported by Russian President grant for young doctorates no. 00-15-99274 and grant RFBR no. 03-01-00304. Mathematics Subject Classification (2000):30E05, 42A82, 44A10     

Deformable orthogonal grids: lemniscates

In this paper we describe a technique, based on complex polynomials, for creating plane regions with a hole and propose a new method to produce an orthogonal grid on it. The thickness of the grid can be easily controlled and the sizes of the cells can be automatically estimated. The grid is automatically adapted to the boundary of the region. We offer parameters for the control of the geometric shape of the region, which depend on the roots of the polynomial and its derivative.     

On estimates of lengths of lemniscates

For any natural number n and any C > 0, we obtain an integral formula for calculating the lengths |L(P _n , C)| of the lemniscates $$L\left( {P_n ,C} \right): = \left\{ {z:\left| {P_n \left( z \right)} \right| = C} \right\}$$ of algebraic polynomials P _n (z):= z ⁿ + c _n?1 z ^n?1 + ... + c ₀ in the complex variable z with complex coefficients c _j , j = 0, ..., n ? 1, and establish the upper bound for the quantities $$\lambda _n : = \sup \left\{ {\left| {L\left( {P_n ,1} \right)} \right|:P_n (z)} \right\},$$ which is currently best for 3 ≤ n ≤ 10¹⁴. We also study the properties of the derivative S′(C) of the area function S(C) of the set {z: |P _n (z)| ≤ C}.     

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.     

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.     

Sequences of polynomials whose zeros lie on fixed lemniscates

For fixed integersk2 we study sequences of polynomialsP _n (z) with the following properties: (i) degP _n ; (ii) the zeros of all theP _n (z) lie on a certain lemniscate withk ₁k foci, one of which is the origin; (iii) theP _n (z) can be cut in such a way that the zeros of the lower part all lie on the unit circle and those of the upper part lie on a lemniscate having the foci in (ii) excluding the origin. Several special cases and examples are considered.     

Bernoulli多项式和Euler多项式的关系

本文给出了 Bernoulli- Euler数之间的关系和 Bernoulli- Euler多项式之间的关系 ,从而深化和补充了有关文献中的相关结果 .     

Bernoulli多项式的积分多项式

首次研究了 Bernoulli多项式的积分多项式 .首先 ,给出这类多项式的定义和基本性质 ;其次 ,建立两类幂和多项式的相互关系 ;最后 ,介绍上述结果在求解自然数幂和公式方面的应用 .     

Euler Pseudoprime Polynomials and Strong Pseudoprime Polynomials

It is usual to emphasize the analogy between the integers and polynomials with coefficients in a finite field, comparing different notions in the two points of view. We introduce a particular rank one Drinfeld module to get an exponentiation for polynomials and then define the notions of Euler pseudoprimes and strong pseudoprimes for polynomials with coefficients in a finite field. As for the integers, we have Solovay–Strassen and Miller–Rabin tests for polynomials.     

On the degree of convergence of lemniscates in finite connected domains

For an arbitrary bounded closed set E in the complex plane with complement Ω of finite connectivity, we study the degree of convergence of the lemniscates in Ω.     

On the size of lemniscates of polynomials in one and several variables

In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set and , for a polynomial of degree . Usually, is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of , in terms of Hausdorff contents, planar Lebesgue measure , or logarithmic capacity cap. Here we normalize and show that cap and are the sharp estimates for the size of . Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in and .

     

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.     

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.     

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

We analyze polynomials P _n that are biorthogonal to exponentials , in the sense that