Electronic ISBN:  9781470401320 
Product Code:  MEMO/115/553.E 
List Price:  $44.00 
MAA Member Price:  $39.60 
AMS Member Price:  $26.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 115; 1995; 117 ppMSC: Primary 14; 32;
A set which can be defined by systems of polynomial inequalities is called semialgebraic. When such a description is possible locally around every point, by means of analytic inequalities varying with the point, the set is called semianalytic. If one single system of strict inequalities is enough, either globally or locally at every point, the set is called basic. The topic of this work is the relationship between these two notions. Namely, Andradas and Ruiz describe and characterize, both algebraically and geometrically, the obstructions for a basic semianalytic set to be basic semialgebraic. Then they describe a special family of obstructions that suffices to recognize whether or not a basic semianalytic set is basic semialgebraic. Finally, they use the preceding results to discuss the effect on basicness of birational transformations.
ReadershipAdvanced graduate students and specialists in real algebra and real geometry.

Table of Contents

Chapters

Introduction

1. Basic and generically basic sets

2. The real spectrum

3. Algebraic and analytic tilde operators

4. Fans and basic sets

5. Algebraic fans and analytic fans

6. Prime cones and valuations

7. Centers of an algebraic fan

8. Henselization of algebraic fans

9. A goingdown theorem, for fans

10. Extension of real valuation rings to the henselization

11. The amalgamation property

12. Algebraic characterization of analytic fans

13. Finite coverings associated to a fan

14. Geometric characterization of analytic fans

15. The fan approximation lemma

16. Analyticity and approximation

17. Analyticity after birational blowingdown


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A set which can be defined by systems of polynomial inequalities is called semialgebraic. When such a description is possible locally around every point, by means of analytic inequalities varying with the point, the set is called semianalytic. If one single system of strict inequalities is enough, either globally or locally at every point, the set is called basic. The topic of this work is the relationship between these two notions. Namely, Andradas and Ruiz describe and characterize, both algebraically and geometrically, the obstructions for a basic semianalytic set to be basic semialgebraic. Then they describe a special family of obstructions that suffices to recognize whether or not a basic semianalytic set is basic semialgebraic. Finally, they use the preceding results to discuss the effect on basicness of birational transformations.
Advanced graduate students and specialists in real algebra and real geometry.

Chapters

Introduction

1. Basic and generically basic sets

2. The real spectrum

3. Algebraic and analytic tilde operators

4. Fans and basic sets

5. Algebraic fans and analytic fans

6. Prime cones and valuations

7. Centers of an algebraic fan

8. Henselization of algebraic fans

9. A goingdown theorem, for fans

10. Extension of real valuation rings to the henselization

11. The amalgamation property

12. Algebraic characterization of analytic fans

13. Finite coverings associated to a fan

14. Geometric characterization of analytic fans

15. The fan approximation lemma

16. Analyticity and approximation

17. Analyticity after birational blowingdown