Abstract
We prove that for a nondegenerate holomorphic map $F=(f(x,y),g(x,y))$ of $\mathbf{C}^2$ to $\mathbf{C}^2$ where $f$ and $g$ are entire functions and $f$ is a transendental one, there exists a ray $J(\theta) = \{(x,y); x = te^{i\theta},y = kte^{i\theta} (0 \leqq t < \infty)\}$ where $k$ is an arbitrarily fixed complex number except some Lebesgue measure zero set and $\theta$ is some real number depending on value $k$, such that $F(x,kx)$, in any open cone in $\mathbf{C}^2$ with vertex $(0,0)$ containing the ray $J(\theta)$, does not omit any algebraic curve with three irreducible components in a general position.
Highlights
Using the theory of normal family, G
We prove that for a nondegenerate holomorphic map F = ( f (x, y), g(x, y)) of C2 to C2 where f and g are entire functions and f is a transendental one, there exists a ray J(θ) = {(x, y); x = teiθ, y = kteiθ (0 t < ∞)} where k is an arbitrarily fixed complex number except some Lebesgue measure zero set and θ is some real number depending on value k, such that F(x, kx), in any open cone in C2 with vertex (0, 0) containing the ray J(θ), does not omit any algebraic curve with three irreducible components in a general position
Definition 3.2 When A is an algebraic curve with three irreducible components in C2 and satisfies the case (1) in the above proposition, we call A in a general position in C2
Summary
Yukinobu Adachi Nishinomiya City, Kurakuen 2-bannchyo, Japan Correspondence: Yukinobu Adachi, Nishinomiya City, Kurakuen 2-bannchyo, Japan. Received: February 19, 2013 Accepted: May 6, 2013 Online Published: June 28, 2013 doi:10.5539/jmr.v5n3p8
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