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Transactions of the American Mathematical Society

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Some consequences of the algebraic nature of $ p(e\sp{i\theta })$

Author: J. R. Quine
Journal: Trans. Amer. Math. Soc. 224 (1976), 437-442
MSC: Primary 30A06
MathSciNet review: 0419743
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Abstract: For polynomial p of degree n, the curve $ p({e^{i\theta }})$ is a closed curve in the complex plane. We show that the image of this curve is a subset of an algebraic curve of degree 2n. Using Bézout's theorem and taking into account imaginary intersections at infinity, we show that if p and q are polynomials of degree m and n respectively, then the curves $ p({e^{i\theta }})$ and $ q({e^{i\theta }})$ intersect at most 2mn times. Finally, let $ {U_k}$ be the set of points w, not on $ p({e^{i\theta }})$, such that $ p(z) - w$ has exactly k roots in $ \vert z\vert < 1$. We prove that if L is a line then $ L \cap {U_k}$ has at most $ n - k + 1$ components in L and in particular $ {U_n}$ is convex.

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Keywords: Polynomials, Bézout's theorem, algebraic curves
Article copyright: © Copyright 1976 American Mathematical Society

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