On the Self-Intersections of the Image of the Unit Circle Under a Polynomial Mapping

Abstract

We prove that ifp is a polynomial of degree n, then with certain exceptions the image of the unit circle under the mapping/; has at most (n -l)2 points of self-intersection.We apply our method to the problem of computing polynomials univalent in W<1. Introduction.Let p(z) be a polynomial in the complex variable z.Definition.We say w is a vertex, or point of self-intersection of the curve p(ex^), 0 = ^_2tt, if there exist zx and z2 distinct with 1^1 = |^2I:= 1 such that p(zx)=p(z2) = w.C. J. Titus [6, p. 60] conjectured that if the degree of p is n, then with certain exceptions the curve p(e"t') has at most (n-l)2 vertices.Our main result is a proof of this conjecture.We also show how the methods used relate to the problem of finding polynomials univalent in |z|<l.Computation of vertices.Let p(z)=a0+axz+-• --\-a"zn.In order to study the vertices of p(el<t'), we study the polynomial G(zx, z2) = (p(z2) -p(zx))/(z2 -zx).We find it convenient following Dieudonné [3] to make a transformation of the variables zx and z2.LetLet ip:A->-B send the pair (r, x) onto the pair (xezB, xe~'°) where 0 = arc cos t, i.e. /=cos 6 and 0_S_-77.Geometrically x is the midpoint of the Presented to the Society, March 31, 1972 under the title On the double points of the image of the unit circle under a polynomial mapping;

Locations

• Proceedings of the American Mathematical Society - View - PDF