Some Consequences of the Algebraic Nature of p(e iθ

Abstract

For polynomial p of degree n, the curve p(e ) 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 Bizout'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(ei0) and q(e0) intersect at most 2mn times. Finally, let Uk be the set of points w, not on p(ei), such that p(z) - w has exactly k roots in Iz I < 1. We prove that if L is a line then L n Uk has at most n - k + 1 components in L and in particular U. is convex. 1. Introduction. Let p be a polynomial of degree n. The curve p(e1), 0 S 0 < 2fr, is a closed curve in the complex plane. We show that the image of the curve is a subset of an algebraic curve of degree 2n, i.e., we can find a poly- nomial h(w, wi) of degree 2n in the variables jointly such that h(p(z), pzi)) = 0 for I z I = 1. This fact used with Bezout's theorem shows that p(e'0) intersects an algebraic curve of degree m at most 2mn times counting multiplicity, where the multiplicity is counted as in algebraic geometry. We will show that if p and q are polynomials of degree m and n respectively, then the curves p(e'0) and q(e'0) intersect at most 2mn times. This will follow again from Be'zout's theorem, but this time certain imaginary intersections at infinity must be taken into account. Finally, let Uk be the set of points w, not on p(ei'), such that p(z) - w has exactly k roots in Iz I < 1. We prove that if L is a line, then L n Uk has at most n - k + 1 components in L and in particular U, is convex. 2. The algebraic nature of p(e'0). Let p be a polynomial of degree n. We look at the closed curve p(e0), 0 S 0 6 2ff. We may write p(ei0) = R(0) + 11(0) where R and I are trigonometric polynomials of degree n, with real coeffi- cients. Let L be the line with equation Ax + By + C = O with A, B and C real. The intersections of p(eW0) with L correspond to the real zeros of AR(0) + BI(0) + C, with the multiplicity of the intersection counted as the multiplicity of the zero. Now AR(0) + 21(0) + C is a trigonometric polynomial of degree n with real coefficients and therefore has at most 2n zeros counting multiplicity