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Register a new userLocal theory of stable polynomials and bounded rational functions of several variables
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We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess non-tangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial $p$, we analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of $p$. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk.
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In this paper, we study the uniqueness of the differential polynomials of entire functions. We prove the following result: Let $f(z)$ be a nonconstant entire function on $mathbb{C}^{n}$ and $g(z)=b_{-1}+sum_{i=0}^{n}b_{i}D^{k_{i}}f(z)$, where $b_{-1}$ and $b_{i} (i=0ldots,n)$ are small meromorphic functions of $f$, $k_{i}geq0 (i=0ldots,n)$ are integers. Let $a_{1}(z) otequivinfty, a_{2}(z) otequivinfty$ be two distinct small meromorphic functions of $f(z)$. If $f(z)$ and $g(z)$ share $a_{1}(z)$ CM, and $a_{2}(z)$ IM. Then either $f(z)equiv g(z)$ or $a_{1}=2a_{1}=2$, $$f(z)equiv e^{2p}-2e^{p}+2,$$ and $$g(z)equiv e^{p},$$ where $p(z)$ is a non-constant entire function on $mathbb{C}^{n}$.
In this paper, we study the uniqueness of meromporphic functions and their difference operators. In particular, We have proved: Let $f$ be a nonconstant entire function on $mathbb{C}^{n}$, let $etain mathbb{C}^{n}$ be a nonzero complex number, and let $a$ and $b$ be two distinct complex numbers in $mathbb{C}^{n}$. If $$varlimsup_{rrightarrowinfty}frac{logT(r,f)}{r}=0,$$ and if $f(z)$ and $(Delta_{eta}^{n}f(z))^{(k)}$ share $a$ CM and share $b$ IM, then $f(z)equiv(Delta_{eta}^{n}f(z))^{(k)}$.
In this note, we use Rouches theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results regarding the geometry of the zeros, poles, and critical points of a rational function. Included is an improvement on a result by Alexander and Walsh regarding the distance from a given zero or pole of a rational function to the nearest critical point.
For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.
For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capacity associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.
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