Considerando germes de hipersuperfícies em \'C POT.n+1\' com conjunto singular de dimensão s, Massey em [14] introduz um conjunto de (s+1) números chamados de números de Lê. Estes números se mostram como a generalização natural do número de Milnor para singularidades isoladas. O principal objetivo deste trabalho é mostrar como estes números são obtidos e que os números de Lê de uma hipersuperfície singular estão relacionados com os números de Lê de uma certa sequência de hipersuperfícies singulares \'X IND.0\',...,\'X IND.s-1\' que se aproxima da singularidade original de tal forma que os conjuntos críticos de seus termos \'X IND.i\' têm dimensão i. Essas relações são dadas pelas fórmulas de Lê-Iomdine. / For any germ of hypersurface in \'C POT. n+1\' with singular set of dimension s, Massey in [14] introduces a set of (s+1) numbers called Lê numbers. These numbers are a natural generalization of the Milnor number for isolated singularity hypersurfaces. The main goal of this work is to show how to obtain these numbers and to show the Lê numbers of a singular hypersurface are related with the the Lê numbers of a sequence of singular hypersurfaces \'X IND.0\',...,\'X IND.s-1\' which approach the original singularity in such a way that the critical set of each \'X IND.i\' has dimension i. These relationship are given by the Lê-Iomdine formulas.

Considerando germes de hipersuperfícies em \'C POT.n+1\' com conjunto singular de dimensão s, Massey em [14] introduz um conjunto de (s+1) números chamados de números de Lê. Estes números se mostram como a generalização natural do número de Milnor para singularidades isoladas. O principal objetivo deste trabalho é mostrar como estes números são obtidos e que os números de Lê de uma hipersuperfície singular estão relacionados com os números de Lê de uma certa sequência de hipersuperfícies singulares \'X IND.0\',...,\'X IND.s-1\' que se aproxima da singularidade original de tal forma que os conjuntos críticos de seus termos \'X IND.i\' têm dimensão i. Essas relações são dadas pelas fórmulas de Lê-Iomdine. / For any germ of hypersurface in \'C POT. n+1\' with singular set of dimension s, Massey in [14] introduces a set of (s+1) numbers called Lê numbers. These numbers are a natural generalization of the Milnor number for isolated singularity hypersurfaces. The main goal of this work is to show how to obtain these numbers and to show the Lê numbers of a singular hypersurface are related with the the Lê numbers of a sequence of singular hypersurfaces \'X IND.0\',...,\'X IND.s-1\' which approach the original singularity in such a way that the critical set of each \'X IND.i\' has dimension i. These relationship are given by the Lê-Iomdine formulas.

Im Rahmen des TERA-Projektes (Turbo Evaluation and Rapid Algorithms) wurde ein neuartiger, hochgradig effizienter probabilistischer Algorithmus zum Lösen polynomialer Gleichungssysteme entwickelt und für den komplexen Fall implementiert. Die Geometrie polarer Varietäten gestattet es, diesen Algorithmus zu einem Verfahren zur Charakterisierung der reellen Lösungsmengen polynomialer Gleichungssysteme zu erweitern. Ziel dieser Arbeit ist es, eine Implementierung dieses Verfahrens zur Bestimmung reeller Lösungen auf eine Klasse von Beispielproblemen anzuwenden. Dabei wurde Wert darauf gelegt, dass diese Beispiele reale, praxisbezogene Anwendungen besitzen. Diese Anforderung ist z.B. für polynomiale Gleichungssysteme erfüllt, die sich aus dem Entwurf von schnellen Wavelet-Transformationen ergeben. Die hier betrachteten Wavelet-Transformationen sollen die praktisch wichtigen Eigenschaften der Orthogonalität und Symmetrie besitzen. Die Konstruktion einer solchen Wavelet-Transformation hängt von endlich vielen reellen Parametern ab. Diese Parameter müssen gewisse polynomiale Gleichungen erfüllen. In der veröffentlichten Literatur zu diesem Thema wurden bisher ausschließlich Beispiele mit endlichen Lösungsmengen behandelt. Zur Berechnung dieser Beispiele war es dabei ausreichend, quadratische Gleichungen in einer oder zwei Variablen zu lösen. Zur Charakterisierung der reellen Lösungsmenge eines polynomialen Gleichungssystems ist es ein erster Schritt, in jeder reellen Zusammenhangskomponente mindestens einen Punkt aufzufinden. Schon dies ist ein intrinsisch schweres Problem. Es stellt sich heraus, dass der Algorithmus des TERA-Projektes zur Lösung dieser Aufgabe bestens geeignet ist und daher eine größere Anzahl von Beispielproblemen lösen kann als die besten kommerziell erhältlichen Lösungsverfahren. / As a result of the TERA-project on Turbo Evaluation and Rapid Algorithms a new type, highly efficient probabilistic algorithm for the solution of systems of polynomial equations was developed and implemented for the complex case. The geometry of polar varieties allows to extend this algorithm to a method for the characterization of the real solution set of systems of polynomial equations. The aim of this work is to apply an implementation of this method for the determination of real solutions to a class of example problems. Special emphasis was placed on the fact that those example problems possess real-life, practical applications. This requirement is satisfied for the systems of polynomial equations that result from the design of fast wavelet transforms. The wavelet transforms considered here shall possess the practical important properties of symmetry and orthogonality. The specification of such a wavelet transform depends on a finite number of real parameters. Those parameters have to obey certain polynomial equations. In the literature published on this topic, only example problems with a finite solution set were presented. For the computation of those examples it was sufficient to solve quadratic equations in one or two variables. To characterize the set of real solutions of a system of polynomial equations it is a first step to find at least one point in each connected component. Already this is an intrinsically hard problem. It turns out that the algorithm of the TERA-project performes very well with this task and is able to solve a larger number of examples than the best known commercial polynomial solvers.