This book provides an exposition of the theory of analytic C-spaces developed by Cartan, Whitney and Tognoli and describes some central results in global real analytic geometry, such as Nullstellensatze and Positivstellensatze, including Forster's global Nullstellensatz for Stein algebras. It emphasizes the central role of Hilbert's 17th Problem in this context, devoting a chapter to the state of the art on this difficult problem. The focus then turns to a class of semianalytic sets defined by countably many global real analytic functions, which is stable under topological operations and satisfies a direct image theorem. A smaller subclass admits a decomposition into irreducible components comparable to that for semialgebraic sets. The last chapter is dedicated to the extension of some of the preceding results to smooth functions and quasi-analytic Denjoy-Carleman functions. The book is addressed to researchers and Ph.D students interested in complex analysis and real analytic geometry.

Autorentext

Francesca Acquistapace was associate professor at the Mathematics Department of Pisa University from 1982 until her retirement in 2017. Previously, from 1974, she was assistant professor at the same department, where she presently has a research contract. She has given Ph.D courses in several universities, including in Madrid, Nagoya, Sapporo and the Poincaré Institute, Paris. Her research is in real analytic geometry, mainly in collaboration with the Spanish team (Andradas, Ruiz, Fernando) and with M. Shiota at Nagoya University. 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.

Klappentext

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.

Inhalt

Introduction

Chapter 1. The class of C-analytic spaces

Chapter 2. More on analytic sets

Chapter 3. Nullstellensätze

Chapter 4. The 17th Hilbert's Problem for real analytic functions

Chapter 5. Analytic inequalities

References