The number of connected components of certain real algebraic curves

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).