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

Full-text PDF

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 M. Marden,*Zeros of self-inversive polynomials*, Proc. Amer. Math. Soc.**3**(1952), 471-475. MR**13**, 938. MR**0047828 (13:938f)****[2]**A. Cohn,*Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise*, Math. Z.**14**(1922), 110-148. MR**1544543****[3]**M. Marden,*Geometry of polynomials*, 2nd ed., Math. Surveys, no. 3, Amer. Math. Soc., Providence, R.I., 1966. MR**37**# 1562. MR**0225972 (37:1562)****[4]**G. Polya and G. Szegö,*Problems and theorems in analysis*, Vol. I, Springer-Verlag, Berlin and New York, 1972. MR**49**#8782. MR**0396134 (53:2)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57D40,
30A08

Retrieve articles in all journals with MSC: 57D40, 30A08

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