Polynomial occurrences in an isolating disk

Abstract

Analytic units and the principal-log boundary primitive compare root multiset occurrences in the original isolating disk.

Let F be analytic near the original closed disk of radius delta>0, with exact order m>=1 at its center c and no other zero. If |P-F|<|F| on the sphere, P has no boundary root and exactly m root occurrences in the disk.

Glue the local unit V to F(z)/(z-c)^m by setting U(c)=V(c). It is analytic and nonzero on the full closed disk, with F(z)=(z-c)^mU(z). Its logarithmic derivative contributes zero to the circle integral by Cauchy-Goursat; the centered factor contributes 2piim.

The boundary quotient P/F lies in the unit ball around 1, hence in the principal-log slit plane. Differentiating log(P/F) gives P’/P-F’/F. The circle primitive theorem makes its integral zero, with continuous boundary integrands supplying integrability.

The polynomial logarithmic derivative is summed over the root multiset with repetitions. Every occurrence inside contributes 2pii; every occurrence outside contributes zero. The nonzero factor 2pii cancels, yielding the original-disk occurrence count used by the source fixed-window clause. This is the classical argument-principle proof of the Rouche disk comparison; it carries no literature originality claim.