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 ""GoingUp"" Theorem for SemiIntegral 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 NonArchimedean Extensions.; 7.5. Orders on Simple Field Extensions; 7.6. Total Orders and Signed Places; 7.7. Existence of Signed Places 

CHAPTER VIII  AFFINE SEMTALGEBRAIC SETS8.1. Introduction and Notation; 8.2. Some Properties of RHJAlgebras; 8.3. Real Curves; 8.4. Signed Places on Function Fields; 8.5. Characterization of NonNegative 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 SemiAlgebraic 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 semialgebraic 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 273277) 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.


Semialgebraischer Raum.

Genre/Form 
Semialgebraische Geometrie.

Form 
Electronic book

ISBN 
9781107360921 

1107360927 

9780511891922 

051189192X 
