On the Nullstellensätze for Stein spaces and C-analytic sets

Original · EN

In this work we prove the real Nullstellensatz for the ring O(X) of analytic functions on a C-analytic set X⊂Rⁿ in terms of the saturation of Łojasiewicz's radical in O(X): The ideal I(Z(a)) of the zero-set Z(a) of an ideal a of O(X) coincides with the saturation √[Ł]a of Łojasiewicz's radical √[Ł]a. If Z(a) has `good properties' concerning Hilbert's 17th Problem, then I(Z(a))=√[r]a where √[r]a stands for the real radical of a. The same holds if we replace √[r]a with the real-analytic radical √[ra]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 O(Rⁿ) and YRⁿ the germ of the support of the coherent sheaf that extends aORⁿ to a suitable complex open neighbourhood of Rⁿ. We study the relationship between a normal primary decomposition of a and the decomposition of YRⁿ as the union of its irreducible components. If a:=p is prime, then I(Z(p))=p if and only if the (complex) dimension of YRⁿ coincides with the (real) dimension of Z(p).