Some consequences of the algebraic nature of 𝑝(𝑒^{𝑖𝜃})

Abstract

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