Paul M. Gauthier, Thomas Ransford, Simon St-Amant, Jérémie Turcotte
We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness can be used to `improve' convergence for sequences of functions.
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We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
The characteristic properties of Artin's denominators in Hilbert's 17th problem are obtained. It is proved that numerators of partial derivative of rational real function from the Nevanlinna class are SOS polynomials.
Necessary and sufficient conditions are obtained under which the numerator of the partial derivative of a rational function holomorphic in open upper poly-halfplane is the sum of squares of polynomials.
In this paper, we give an explicit criterion when a rational holomorphic map between balls is equivalent to a polynomial holomorphic map. Making use of this criterion, we show that any proper rational holomorphic map from B^2 into B^N of degree two is equivalent to a polynomial holomorphic map; we also construct rational holomorphic maps of degree 3 that are almost linear but are not equivalent to polynomial holomorphic maps.
It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that is homeomorphic to $K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in ${\mathbb C}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown als…
This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+zt^{r}$, where the zeros of $P$ are positive and real. We show that every member of a family of such generating functions - parametrized by the degree of $P$ and $r$ - gives rise to a sequence of polynomials $\{H_{m}(z)\}_{m=0}^{\infty}$ that is eventually hyperbolic. Moreover, when $P(0)>0$ the real zeros of the polynomials $H_{m}(z)$ form a dense subset of an interval $I\subset\mathbb{R}^{+}$, whose length depends on the particular values…
This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions gives rise to a sequence of polynomials $\{P_{m}(z)\}_{m=0}^{\infty}$ that is eventually hyperbolic. Moreover, the real zeros of the polynomials $P_{m}(z)$ form a dense subset of an interval $I\subset\mathbb{R}^{+}$, whose length depends on the particular values of the parameters in the generating function.
In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein-Erdélyi inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.
J.Ritt has investigated the structure of complex polynomials with respect to superposition. In particular, he listed all the polynomials admitting different double decompositions into indecomposable polynomials. The analogues of Ritt theory for rational functions were constructed just for several particular classes of the said functions, say for Laurent polynomials (F.Pakovich). In this note we describe a certain class of double decompositions for rational functions. Essentially, described below rational functions were discovered by E.I.Zolotarev in 1877 as a solution of certain optimization p…
We solved the problem of the best rational approximation of the Bergman kernels on the unit circle of the complex plane in the quadratic and uniform metrics.
We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.
Stable polynomials and bounded rational functions in the unit ball
研究单位球内无零点的多项式及其对有理函数有界性的贡献，通过局部描述和高维特征分析，探讨有理函数在球上的有界性问题。
It is shown that rational and polynomial convexity of totally real submanifolds is in general unstable under perturbations that are $C^α$-small for any Hölder exponent $α<1$. This complements the result of Løw and Wold that these properties are $C^1$-stable.
In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(γ, 1! (2a+1)ζ(2), 2!ζ(3),..., (m-1)!(a+1+(-1)^ma)ζ(m)),$ $m=1,2,...,a,$ $a\in{\mathbb N}.$
We establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1$,...,$f_n$ and $n'$ functions $g_1,...,g_{n'}$ in $\mathcal{C}^\infty(M)$ so that every element of $\mathcal{C}^k(M)$ can be approximated by rational combinations of $f_1,...,f_n$ and polynomial combinations of $g_1,...,g_{n'}$. As an application, we show that the optimal value of $n$ and $n'$ for all manifolds of dimension $m$ is [3m/2], when $k\geq 1$ and $m\geq 2$.
Polynomial and rational convergence rates for Laplace problems on planar domains
本文研究平面非凸解析区域上拉普拉斯问题的最小二乘展开，证明多项式与有理逼近的收敛速率可能存在巨大差异，并综合Schwarz函数、位势理论与共形映射拥挤理论给出证明。
Local theory of stable polynomials and bounded rational functions of several variables
本文基于齐次分解、Puiseux展开和传递函数实现提供稳定多项式的局部理论，借此证明多圆盘有界有理函数的边界非切向极限，并刻画双上半平面分子理想与双圆盘可积性。
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