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 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)
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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