Acquistapace, FrancescaBroglia, FabrizioFernando Galván, José Francisco2023-06-182023-06-1820160002-994710.1090/tran/6436https://hdl.handle.net/20.500.14352/24540In 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(Ƶ(a)) of the zero-set Ƶ(a) of an ideal a of O(X) coincides with the saturation (Formula presented) of Łojasiewicz’s radical (Formula presented). If Ƶ(a) has ‘good properties’ concerning Hilbert’s 17th Problem, then I(Ƶ(a)) = (Formula presented) where (Formula presented) stands for the real radical of a. The same holds if we replace (Formula presented) with the real-analytic radical (Formula presented) of a, which is a natural generalization 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 neighborhood 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(Ƶ(p)) = p if and only if the (complex) dimension of YRn coincides with the (real) dimension of Ƶ(p).engjournal articlehttp://www.ams.org/journals/tran/2016-368-06/S0002-9947-2015-06436-8/S0002-9947-2015-06436-8.pdfhttp://www.ams.org/restricted access517.5NullstellensatzStein spaceClosed idealRadicalReal NullstellensatzC-analytic setSaturated idealLojasiewicz’s radicalConvex idealH-setsHa-setReal idealReal radicalReal-analytic idealReal-analytic radicalQuasi-real ideal.Funciones (Matemáticas)1202 Análisis y Análisis Funcional