Topics in Global Real Analytic Geometry (Springer Monographs in Mathematics)

✍ Scribed by Francesca Acquistapace, Fabrizio Broglia, José F. Fernando

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 285
• Category: Library

✦ Synopsis

In the first two chapters we review the theory developped by Cartan, Whitney and Tognoli. Then Nullstellensatz is proved both for Stein algebras and for the algebra of real analytic functions on a C-analytic space. Here we find a relation between real Nullstellensatz and seventeenth Hilbert’s problem for positive semidefinite analytic functions. Namely, a positive answer to Hilbert’s problem implies a solution for the real Nullstellensatz more similar to the one for real polinomials. A chapter is devoted to the state of the art on this problem that is far from a complete answer.

In the last chapter we deal with inequalities. We describe a class of semianalytic sets defined by countably many global real analytic functions that is stable under topological properties and under proper holomorphic maps between Stein spaces, that is, verifies a direct image theorem. A smaller class admits also a decomposition into irreducible components as it happens for semialgebraic sets. During the redaction some proofs have been simplified with respect to the original ones.

✦ Table of Contents

Preface
Acknowledgments
Contents
1 The Class of C-Analytic Spaces
1.1 Complex Analytic Spaces
1.1.1 Local Properties
1.1.1.1 Regular Points of a Reduced Complex Analytic Space
1.1.1.2 Zariski's Tangent Space
1.1.2 Stein Spaces
1.1.2.1 Cartan's Theorems A and B and Direct Limits
1.1.2.2 Characterizations of Stein Spaces
1.2 Real Analytic Spaces
1.2.1 Complexification
1.2.1.1 The Local Case
1.2.1.2 A Gluing Lemma
1.2.1.3 The Global Case
1.2.2 Anti-Involutions and Real Parts
1.2.3 Real Structure of a Complex Analytic Set
1.2.4 Real Analytic Subspaces of Rn
1.2.5 Well Reduced Structure
Bibliographic and Historical Notes
2 More on Analytic Sets
2.1 Irreducible Components
2.1.1 Irreducible Components of a Complex Analytic Set
2.1.2 Irreducible Components of a C-Analytic Set
2.2 Normalization
2.2.1 The Normalization Sheaf widecheckOo
2.2.2 Properties of the Normalization
2.2.3 The Real Case
2.3 Divisors in C-Analytic Sets
2.3.1 Multiplicities
2.3.2 Divisors
2.3.3 Locally Principal Divisors
Bibliographic and Historical Notes
3 Nullstellensätze
3.1 Nullstellensatz for Stein Spaces
3.1.1 Complex Stein Algebras as Fréchet Spaces
3.1.2 Nullstellensatz for Closed Primary Ideals
3.1.3 Primary Decomposition
3.1.4 Nullstellensatz for Closed Ideals
3.2 The Real Case
3.2.1 A Global Łojasiewicz's Inequality
3.2.2 Nullstellensatz for General Ideals
3.2.3 Real Radical and Łojasiewicz Radical
Bibliographic and Historical Notes
4 Hilbert's 17th Problem for Real Analytic Functions
4.1 Artin–Schreier Theory and the Local Case
4.2 The Low-Dimensional Global Case
4.2.1 Dimension 1
4.2.2 Dimension 2
4.3 The Pythagoras Number for Curves and Surfaces
4.4 Excellent Rings
4.5 Compact Zerosets
4.6 Infinite Sums of Squares
4.7 Countably Many Compact Sets
4.7.1 Controlling Denominators
4.7.2 Globalization of Sums of Squares
4.7.3 Consequences
4.7.4 Consequences on Pythagoras Numbers
4.8 The Discrete Case
4.9 Pfister's Trick
4.9.1 Globally Principal Ideal Sheafs
4.9.2 Sheaf-Products
4.9.3 Sheaf-Extension
4.9.4 Infinite Sums of Squares of Analytic Functions
4.9.5 Bounded Analytic Pfister's Formula
4.9.5.1 Pfister's Multiplicative Formula
4.9.5.2 Pfister Bundles
4.9.6 Unbounded Analytic Pfister's Formula
4.9.6.1 Hilbert Spaces, Holomorphic Functions and Vector Bundles
4.9.6.2 Pfister's Multiplicative Formula
4.9.7 Sums of Countably Many Squares
4.9.8 Applications to Hilbert's 17th Problem
4.9.8.1 Irreducible Factors
4.9.8.2 Applications
4.10 Examples
Bibliographic and Historical Notes
5 Analytic Inequalities
5.1 Global Semianalytic Sets
5.1.1 Global Semianalytic Subsets of an Analytic Curve
5.1.2 Global Semianalytic Subsets of 2dimensionalManifolds
5.1.3 General Dimension
5.2 Strict Positivstellensatz
5.3 C-Semianalytic Sets
5.3.1 The Direct Image Theorem
5.3.2 Subanalytic Sets as Proper Imagesof C-Semianalytic Sets
5.3.3 Local Extrema of a Real Analytic Function
5.3.4 Points at which a C-Analytic Set is not Coherent
5.4 Amenable C-Semianalytic Sets and Irreducible Components
5.4.1 Characterization of Amenable C-Semianalytic Sets
5.4.2 Images of Amenable C-Semianalytic Sets Under Proper Holomorphic Maps
5.4.3 Tameness-Algorithm for C-Semianalytic Sets
5.4.4 Normalization and Irreducibility
5.4.5 Irreducible Components of an Amenable C-Semianalytic Set
Bibliographic and Historical Notes
6 Other Structures
6.1 The Algebra OoRn as a Subalgebra of EnvRn
6.1.1 Germs and Arcs
6.1.2 Global Properties
6.2 The Algebra EnvRn
6.2.1 Positivstellensatz
6.2.2 Nullstellensatz
6.2.2.1 Łojasiewicz Ideals
6.2.2.2 Weakly Łojasiewicz Ideals
6.2.2.3 Some Consequences
6.3 Quasi-Analytic Denjoy–Carleman Algebras
Bibliographic and Historical Notes
References
Index