scholarly journals The Distribution of Zeros and Poles of Asymptotically Extremal Rational Functions for Zolotarev's Problem

2001 ◽  
Vol 110 (1) ◽  
pp. 88-108 ◽  
Author(s):  
A.L Levin ◽  
E.B Saff
Keyword(s):  
Rational Functions ◽  
Distribution Of Zeros ◽  
Zeros And Poles

scholarly journals Salem Numbers and Pisot Numbers via Interlacing

2012 ◽  
Vol 64 (2) ◽  
pp. 345-367 ◽  
Author(s):  
James McKee ◽  
Chris Smyth
Keyword(s):  
Unit Circle ◽  
Rational Functions ◽  
General Construction ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Limit Points ◽  
Zeros And Poles

Abstract We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the “obvious” limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

Research on Distribution of Zeros and Poles for Plant Family with Parametric Uncertainty

2012 ◽  
Vol 433-440 ◽  
pp. 2367-2371
Author(s):  
Ze Wei Zheng
Keyword(s):  
Parametric Uncertainty ◽  
Distribution Of Zeros ◽  
Plant Family ◽  
The Relationship ◽  
Roots Of A Polynomial ◽  
Zeros And Poles

This paper considers the distribution of zeros and poles of plant family with parametric uncertainty. The relationship between the coefficients and the roots of a polynomial is also discussed. The computation of the distributions of spherical plant family is illustrated in this paper.

Construction of rational functions on a curve

1970 ◽  
Vol 68 (1) ◽  
pp. 105-123 ◽  
Author(s):  
J. Coates
Keyword(s):  
Diophantine Equations ◽  
Rational Functions ◽  
Subsequent Paper ◽  
Complete Proof ◽  
Integer Points ◽  
Detailed Proof ◽  
Explicit Bounds ◽  
Zeros And Poles

1. Introduction. In the study of diophantine equations in two variables, it is often necessary to consider rational functions on a curve with prescribed zeros and poles. Although it is well known that such functions can, in principle, always be effectively constructed, the detailed proof does not appear to have been given. The purpose of the present paper is to give the complete proof of such a construction. Our method, and the statement of our results, are motivated by the applications to diophantine equations which we have in mind. In particular, our results will play an important role in a subsequent paper (1), in which explicit bounds will be established for the integer points on any curve of genus 1.

scholarly journals On the Distribution of Zeros and Poles of Rational Approximants on Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
V. V. Andrievskii ◽  
H.-P. Blatt ◽  
R. K. Kovacheva
Keyword(s):  
Asymptotic Distribution ◽  
Polynomial Approximation ◽  
Additional Assumption ◽  
Equilibrium Measure ◽  
Distribution Of Zeros ◽  
Rational Approximants ◽  
Bounded Degree ◽  
Asymptotic Distribution Of Zeros ◽  
Best Rational Approximants ◽  
Zeros And Poles

The distribution of zeros and poles of best rational approximants is well understood for the functions , . If is not holomorphic on , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, -values, and poles of best real rational approximants of degree at most to a function that is real-valued, but not holomorphic on . Generalizations to the lower half of the Walsh table are indicated.

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