Die Anzahl der Nullstellen eines Polynoms in Gebieten mit stückweise rationalen Randkurven

Abstract

We are dealing with the problem of calculating the numberN of zeros of a polynomialw(z) in certain regions Γ of the complex plane. If Γ is a halfplane or a disc, there exist many well-known solutions of the problem. If Γ is of a more general type, there is only a paper ofSherman4) for the case, that Γ is the intersection of two halfplanes. In this paper we give the solution for the case that the region Γ is simply connected and the boundary of Γ is composed by a finite number of pieces of rational curves. Our method generalizes the method ofWielandt3) for the halfplane and the method ofSherman4). The core of our paper is the way of computing the variation of argw, ifw runs along a piece of a rational curve. This computation is based on a lemma about generalized Sturm sequences.