Overview
- Addresses researchers and Ph.D students interested in complex analysis and real analytic geometry
- Provides the first book treatment of fundamental results on Stein algebras and real analytic spaces
- Offers a perspective on real analytic geometry from Whitney and Cartan to the present day
Part of the book series: Springer Monographs in Mathematics (SMM)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
Table of contents (6 chapters)
Keywords
- Complex analytic spaces
- Real analytic spaces
- Irreducible components.
- Normalization
- Divisors in C-analytic Sets
- Nullstellensatz for Stein spaces and real analytic spaces
- Artin-Schreier theory
- Pytagoras number
- Excellent rings
- Infinite sums of squares
- Pfister's forms and matrices
- Global semianalytic sets
- Strict Positivstellensatz
- C-semianalytic sets
- Amenable C-semianalytic sets and irreducible components
About this book
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. Duringthe redaction some proofs have been simplified with respect to the original ones.
Reviews
“This noteworthy book fulfills the goal of giving an excellently well written account of the present state of a number of relevant topics in the field of Real Analytic Geometry.” (José Javier Etayo, zbMATH 1495.14001, 2022)
Authors and Affiliations
About the authors
Fabrizio Broglia was full professor at the Mathematics Department of Pisa University from 2001 until his retirement in 2018. Previously he was assistant and associate professor at the same Department, where he presently has a research contract. He was director of the Ph.D school of Science from 2002 until 2016. He was responsible in Italy for two European networks in Real Algebraic and Analytic Geometry (RAAG). His research deals with real analytic geometry, in collaboration with many colleagues, in particular the Spanish team.
José F. Fernando has been Professor at the Universidad Complutense de Madrid since February 2021. He has actively worked in Real Algebraic and Analytic Geometry (RAAG) with groups in Spain (Baro, Gamboa, Ruiz, Ueno), Duisburg-Konstanz (Scheiderer), Pisa (Acquistapace–Broglia), Rennes (Fichou–Quarez), and Trento (Ghiloni). He has established a strong collaboration and friendship with the Pisa RAAG group since 2003.
Bibliographic Information
Book Title: Topics in Global Real Analytic Geometry
Authors: Francesca Acquistapace, Fabrizio Broglia, José F. Fernando
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-030-96666-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-96665-2Published: 08 June 2022
Softcover ISBN: 978-3-030-96668-3Published: 09 June 2023
eBook ISBN: 978-3-030-96666-9Published: 07 June 2022
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XVII, 273
Number of Illustrations: 3 b/w illustrations
Topics: Algebraic Geometry