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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On univalent polynomials


Author: J. R. Quine
Journal: Proc. Amer. Math. Soc. 57 (1976), 75-78
MSC: Primary 30A34; Secondary 30A06
DOI: https://doi.org/10.1090/S0002-9939-1976-0402024-3
MathSciNet review: 0402024
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Abstract: We define $ {V_n} \subseteq {{\mathbf{C}}^{n - 1}}$ to be the set of $ (n - 1)$-tuples $ ({a_2}, \ldots ,{a_n})$ such that the polynomial $ p(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n}$ is univalent, i.e., one-to-one in $ \vert z\vert < 1$. In this paper we construct a real polynomial $ h$ of degree $ 4(2{(n - 1)^2} - 1)(n - 1)$ such that if $ ({a_2}, \ldots ,{a_n})$ is in the boundary of $ {V_n}$ then $ h(\operatorname{Re} {a_2},\operatorname{Im} {a_2}, \ldots ,\operatorname{Re} {a_n},\operatorname{Im} {a_n}) = 0$. This shows that the boundary of $ {V_n}$ is a subset of an algebraic submanifold of $ {R^{2(n - 1)}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0402024-3
Keywords: Univalent, polynomials
Article copyright: © Copyright 1976 American Mathematical Society

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