Some consequences of the algebraic nature of $p(e^{i\theta })$

Abstract:

For polynomial p of degree n, the curve $p({e^{i\theta }})$ 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 2n. 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 $p({e^{i\theta }})$ and $q({e^{i\theta }})$ intersect at most 2mn times. Finally, let ${U_k}$ be the set of points w, not on $p({e^{i\theta }})$, such that $p(z) - w$ has exactly k roots in $|z| < 1$. We prove that if L is a line then $L \cap {U_k}$ has at most $n - k + 1$ components in L and in particular ${U_n}$ is convex.