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.
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.
