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 | References | Similar Articles | Additional Information

Abstract: Let be a immersion of the circle. Let be the number of zeros of and suppose for ; then where is the tangent winding number, and . This generalizes the theorem of Cohn that if is a self-inversive polynomial, the number of zeros of in is the same as the number of zeros of in . For , this is a topological generalization of Lucas' theorem. We show how represents a generalization of the notion of the winding number of about 0.

**[1]**F. F. Bonsall and Morris Marden,*Zeros of self-inversive polynomials*, Proc. Amer. Math. Soc.**3**(1952), 471–475. MR**0047828**, https://doi.org/10.1090/S0002-9939-1952-0047828-8**[2]**A. Cohn,*Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise*, Math. Z.**14**(1922), no. 1, 110–148 (German). MR**1544543**, https://doi.org/10.1007/BF01215894**[3]**Morris Marden,*Geometry of polynomials*, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR**0225972****[4]**G. Pólya and G. Szegő,*Problems and theorems in analysis. Vol. II*, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry; Die Grundlehren der Mathematischen Wissenschaften, Band 216. MR**0396134**

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Additional Information

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