The Nullstellensatz for real coherent analytic surfaces

Trans. Amer. Math. Soc.

Broglia

2015

Self Cite

Abstract. In this work we prove the real Nullstellensatz for the ring OpXq of analytic functions on a C-analytic set X Ă R n in terms of the saturation of Lojasiewicz's radical in OpXq: The a of a, which is a natural generalisation of the real radical ideal in the C-analytic setting. We revisit the classical results concerning (Hilbert's) Nullstellensatz in the framework of (complex) Stein spaces.Let a be a saturated ideal of OpR n q and Y R n the germ of the support of the coherent sheaf that extends aO R n to a suitable complex open neighbourhood of R n . We study the relationship between a normal primary decomposition of a and the decomposition of Y R n as the union of its irreducible components. If a :" p is prime, then IpZppqq " p if and only if the (complex) dimension of Y R n coincides with the (real) dimension of Zppq.

Section: Introductionmentioning

confidence: 99%

On the Nullstellensätze for Stein spaces and 𝐶-analytic sets

Trans. Amer. Math. Soc.

Broglia

2015

Self Cite

Topics in Global Real Analytic Geometry

Springer Monographs in Mathematics

Broglia

2022

This series publishes advanced monographs giving well-written presentations of the "state-of-the-art" in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

On the irreducible components of globally defined semianalytic sets

2016

Math. Z.

In this work we present the concept of amenable C-semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable Csemianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets.