The number of connected components of certain real algebraic curves (Q5945163)

Summary: For an integer \(n\geq 2\), let \(p(z)= \prod^n_{k=1} (z-\alpha_k)\) and \(q(z)= \prod^n_{k=1} (z-\beta_k)\), where \(\alpha_k\), \(\beta_k\) are real. We find the number of connected components of the real algebraic curve \(\{(x,y)\in \mathbb{R}^2:|p(x+ iy)|-|q(x+ iy)|= 0\}\) for some \(\alpha_k\) and \(\beta_k\). 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 \(p(z)+ q(z)\).