José Francisco Fernando Galván

ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

In this work we define a semialgebraic set S Rn to be irreducible if the noetherian ring of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring : Substitution Theorem, Positivstellens¨atze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

On the complements of 3-dimensional convex polyhedra as polynomial images of R3

We prove that the complement

On the semialgebraic Stone-Čech compactification of a semialgebraic set

{\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}

On the nullstellensätze for stein spaces and C-analytic sets.

of a 3-dimensional convex polyhedron

On globally defined semianalytic sets

{\mathcal K}\subset{\mathbb R}^3

On Hilbert's 17th problem and Pfister's multiplicative formulae for the ring of real analytic functions

and its closure

Algebra lineal (2ª ed.)

\overline{{\mathcal S}}

ON THE HILBERT 17TH PROBLEM FOR GLOBAL ANALYTIC FUNCTIONS

are polynomial images of

Factorización en algunos subanillos de C

{\mathbb R}^3

Sobre las propiedades de la frontera de las imágenes polinómicas y regulares de Rn

. The former techniques cannot be extended in general to represent such semialgebraic sets