Conjuntos semialgébricos e o teorema da finitude topológica

• Ano de defesa: 2022
• Autor(a) principal: Tanure, André Dantas
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Palavras-chave em Português: Geometria algébrica real; Conjuntos semialgébricos; Real algebraic geometry; Semi-algebraic sets
• Link de acesso: http://www.repositorio.ufc.br/handle/riufc/68082

We present the definition of a semialgebraic subset of Rn, that is a subset defined by a boolean condition of polynomial conditions. We show the first structure theorem for semialgebraic sets, The Cylindrical Decomposition Theorem and explore some of its consequences, in particular the Tarski-Seidenberg Theorem. Then we prove the second structure theorem, The Stratifi cation Theorem. After that, we explore some of its consequences, in particular the concept of dimension, and a version of Sard’s Theorem for semialgebraic mappings. We then present the concepts of Cell Decomposition and Triangulation of a set, and we prove that every compact semialgebraic set admits both a cell decomposition and a triangulation. We then enunciate the Local Triviality Theorem and present two applications: The Theorem on the Finiteness of Topological Types of semialgebraic sets, and the Local Conical Structure Lemma.