Local WEIERSTRASS Theory

Abstract

All local function theory originates from the famous Weierstrass Preparation Theorem which “prepares us so well” for all further discussions. WEIERSTRASS himself called his theorem “Vorbereitungssatz” (cf. Math. Werke 2, p.135), he writes there in a footnote: “Diesen Satz habe ich seit dem Jahre 1860 wiederholt in meinen Universitäts-Vorlesungen vorgetragen.” The Preparation Theorem expresses the fundamental fact that the zero set of a holomorphic function g displays, at least locally in suitable coordinates, an “algebraic” and hence “finite” character. This is the reason why finite holomorphic maps nowadays are the most important tool in local function theory.