International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 11, Pages 693-701
doi:10.1155/S0161171201010481
The number of connected components of certain real algebraic curves
School of Mathematical Sciences, Seoul National University, Seoul 151-742, South Korea
Received 7 June 2000; Revised 7 August 2000
Copyright © 2001 Seon-Hong Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
For an integer n≥2, let p(z)=∏k=1n(z−αk) and q(z)=∏k=1n(z−βk), where αk,βk are real. We find the number of connected components of the real
algebraic curve {(x,y)∈ℝ2:|p(x+iy)|−|q(x+iy)|=0} for some αk and βk. Moreover, in these cases, we show that each connected component contains zeros of p(z)+q(z), and we investigate the locus of zeros of p(z)+q(z).