Some consequences of the algebraic nature of

Author:
J. R. Quine

Journal:
Trans. Amer. Math. Soc. **224** (1976), 437-442

MSC:
Primary 30A06

DOI:
https://doi.org/10.1090/S0002-9947-1976-0419743-X

MathSciNet review:
0419743

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Abstract | References | Similar Articles | Additional Information

Abstract: For polynomial *p* of degree *n*, the curve is a closed curve in the complex plane. We show that the image of this curve is a subset of an algebraic curve of degree 2*n*. Using Bézout's theorem and taking into account imaginary intersections at infinity, we show that if *p* and *q* are polynomials of degree *m* and *n* respectively, then the curves and intersect at most 2*mn* times. Finally, let be the set of points *w*, not on , such that has exactly *k* roots in . We prove that if *L* is a line then has at most components in *L* and in particular is convex.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0419743-X

Keywords:
Polynomials,
Bézout's theorem,
algebraic curves

Article copyright:
© Copyright 1976
American Mathematical Society