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Proceedings of the American Mathematical Society

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

 

 

On positively turning immersions


Author: J. R. Quine
Journal: Proc. Amer. Math. Soc. 61 (1976), 69-72
MSC: Primary 57D40; Secondary 30A08
DOI: https://doi.org/10.1090/S0002-9939-1976-0431210-1
MathSciNet review: 0431210
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Abstract: Let $ \gamma :{S^1} \to {\mathbf{C}}$ be a $ {C^2}$ immersion of the circle. Let $ k$ be the number of zeros of $ \gamma $ and suppose $ d\arg \gamma ({e^{i\theta }})/d\theta > 0$ for $ \gamma ({e^{i\theta }}) \ne 0$; then $ \operatorname{twn} \gamma = k/2 + {(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma $ where $ \gamma $ is the tangent winding number, and $ A = {S^1} - {\gamma ^{ - 1}}(0)$. This generalizes the theorem of Cohn that if $ p$ is a self-inversive polynomial, the number of zeros of $ p'$ in $ \vert z\vert > 1$ is the same as the number of zeros of $ p$ in $ \vert z\vert > 1$. For $ k = 0$, this is a topological generalization of Lucas' theorem. We show how $ {(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma $ represents a generalization of the notion of the winding number of $ \gamma $ about 0.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0431210-1
Keywords: Tangent winding number, Cohn's theorem, Lucas' theorem, self-inverse polynomials, immersions
Article copyright: © Copyright 1976 American Mathematical Society