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Author Brumfiel, Gregory W

Title Partially ordered rings and semi-algebraic geometry / Gregory W. Brumfiel
Published Cambridge [England] ; New York : Cambridge University Press, 1979


Description 1 online resource (280 pages) : illustrations
Series London Mathematical Society lecture note series ; 37
London Mathematical Society lecture note series ; 37.
Contents 5.7. The Structure Sheaf, II -- The Sheaf Axioms for Basic Open Sets5.8. The Structure Sheaf, III -- Definition; CHAPTER VI -- POLYNOMIALS; 6.1. Polynomials as Functions; 6.2. Adjoining Roots; 6.3. A Universal Bound on the Roots of Polynomials; 6.4. A ""Going-Up"" Theorem for Semi-Integral Extensions; CHAPTER VII -- ORDERED FIELDS; 7.1. Basic Results; 7.2. Function Theoretic Properties of Polynomials; 7.3. Sturm's Theorem; 7.4. Dedekind Cuts; Archimedean and Non-Archimedean Extensions.; 7.5. Orders on Simple Field Extensions; 7.6. Total Orders and Signed Places; 7.7. Existence of Signed Places
CHAPTER VIII -- AFFINE SEMT-ALGEBRAIC SETS8.1. Introduction and Notation; 8.2. Some Properties of RHJ-Algebras; 8.3. Real Curves; 8.4. Signed Places on Function Fields; 8.5. Characterization of Non-Negative Functions; 8.6. Derived Orders; 8.7. A Preliminary Inverse Function Theorem; 8.8. Algebraic Simple Points, Dimension, Codimension and Rank; 8.9. Stratification of Semi-Algebraic Sets; 8.10. Krull Dimension; 8.11. Orders on Function Fields; 8.12. Discussion of Total Orders on R(x, y); 8.13. Brief Discussion of Structure Sheaves; I -- The rational structure sheaf
II -- The semi-algebraic structure sheaf
Summary The purpose of this unique book is to establish purely algebraic foundations for the development of certain parts of topology. Some topologists seek to understand geometric properties of solutions to finite systems of equations or inequalities and configurations which in some sense actually occur in the real world. Others study spaces constructed more abstractly using infinite limit processes. Their goal is to determine just how similar or different these abstract spaces are from those which are finitely described. However, as topology is usually taught, even the first, more concrete type of problem is approached using the language and methods of the second type. Professor Brumfiel's thesis is that this is unnecessary and, in fact, misleading philosophically. He develops a type of algebra, partially ordered rings, in which it makes sense to talk about solutions of equations and inequalities and to compare geometrically the resulting spaces. The importance of this approach is primarily that it clarifies the sort of geometrical questions one wants to ask and answer about those spaces which might have physical significance
Bibliography Includes bibliographical references (pages 273-277) and index
Notes Print version record
Subject Commutative rings.
Categories (Mathematics)
Geometry, Algebraic.
MATHEMATICS -- Algebra -- Intermediate.
Geometry, Algebraic
Categories (Mathematics)
Commutative rings
Geordneter Ring
Semialgebraischer Raum
Algebraïsche meetkunde.
Semi-algebraischer Raum.
Genre/Form Semi-algebraische Geometrie.
Form Electronic book
ISBN 9781107360921