Ruiz Sancho, Jesús María

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First Name
Jesús María
Last Name
Ruiz Sancho
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Álgebra, Geometría y Topología
Geometría y Topología
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Now showing 1 - 10 of 72
  • Publication
    Sobre compactificaciones de Wallman-Frink de espacios discretos
    (Real Sociedad Matemática Española;Consejo Superior de Investigaciones Científicas. Instituto "Jorge Juan" de Matemáticas, 1980) Alonso García, María Emilia; Etayo Gordejuela, J. Javier; Gamboa, J. M.; Ruiz Sancho, Jesús María
    Dado un espacio T3α (X,T), es posible obtener una compactificación T2 del mismo, mediante ultrafiltros asociados a ciertas bases distinguidas de cerrados de (X,T) (Frink [4]). Se plantea así el problema siguiente: ¿Puede obtenerse toda compactificación T2 de (X,T) por este método? Desde el año 1964 en que Frink lo planteó, este interrogante ha tenido respuestas afirmativas parciales. Sin embargo, la solución definitiva es negativa.
  • Publication
    Sobre álgebras de Nash
    (Universitat Autònoma de Barcelona, 1980) Ruiz Sancho, Jesús María
    We obtain some results (a Nullstellensatz, a specialization theorem, `à la E. Artin') for Nash algebras with an algebraic method based on M. Artin's theorem (and easily generalizable to the analytic case) notably simplifying known proofs
  • Publication
    On the real spectrum of a ring of global analytic functions
    (Institut de Recherche Mathématique de Rennes, 1986) Ruiz Sancho, Jesús María
    We present here some applications of the theory of real spectra of excellent rings to the ring of global analytic functions on a compact real analytic manifold. Section 1 contains the facts of the theory that shall be used in the sequel. Section 2 describes the good relationship between global semianalytic subsets of the manifold and constructible subsets of the real spectrum of the ring of global analytic functions. This leads to the solution of Hilbert's 17th problem, to the real Nullstellensatz and to the finiteness theorems, all in this global analytic setting. Finally, Section 3 gives a quick overview on several questions related to connectedness, either of constructible sets or of global semianalytic sets
  • Publication
    On the pythagoras numbers of real analytic set germs.
    (Société Mathématique de France, 2005) Fernando Galván, José Francisco; Ruiz Sancho, Jesús María
    We Show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.
  • Publication
    Cônes locaux et complétions
    (Elsevier, 1986-01-14) Ruiz Sancho, Jesús María
    Let A be a local ring; let  denote the completion of A and Spec r A,Spec R  the real spectra of A and Â,respectively. The author studies the fibers of the canonical morphism Spec r  → Spec r A, for an excellent ring A, and computes its images. Two applications are given: an existence theorem for real valuations in excellent rings, and an Artin-Lang homomorphism theorem for global analytic functions
  • Publication
    On the connected components of a global semianalytic set
    (Walter de Gruyter & co, 1988) Ruiz Sancho, Jesús María
    Let M be a real analytic manifold and O(M) its ring of global analytic functions. A global semianalytic subset of M is any set Z of the form Z=⋃ i=0 r {x∈M:fi1(x)>0,⋯,fis(x)>0,gi(x)=0}, (1.1), where fij,gi∈O(M). This imitates the definitions of semialgebraic sets and semianalytic germs, and gives rise to the same old basic problems: Can the gi's in (1.1) be omitted if Z is open? Is the closure of Z global semianalytic when Z itself is? And the connected components of Z? In an earlier paper [in Algèbre, 84–95, Univ. Rennes I, Rennes, 1986] we showed that this is possible for the first two questions in case M is compact: our method relied upon the theory of the real spectrum. In this note we deal with the third question and prove Theorem 1.2: Let Z be a global semianalytic subset of a real analytic manifold M. Assume that Z is relatively compact. Then the connected components of Z are global semianalytic subsets of M. For the proof, we use again the real spectrum, plus the solution by Ch. Rotthaus of M. Artin's conjecture on the approximation property of excellent rings [Rotthaus, Invent. Math. 88 (1987), no. 1, 39–63].
  • Publication
    A characterization of sums of 2nth powers of global meromorphic functions
    (American Mathematical Society, 1990-08) Ruiz Sancho, Jesús María
    Hilbert's 17th Problem asked the following. Let f(x1,⋯,xn) be a real polynomial which for all real values α1,⋯,αn satisfies f(α1,⋯,αn)≥0. Is it true that f=∑(gi/hi)2 for polynomials gi,hi∈R[x1,⋯,xn]? (It was known that f=∑(gi)2, gi∈R[x1,⋯,xn], is not always true.) Artin gave a positive answer to this problem. In this paper the author proves an analogous theorem with f replaced by a real-analytic function on a compact analytic subvariety X of a real analytic manifold, the rational functions gi/hi replaced by meromorphic functions, the power 2 replaced by 2n and the condition "f(α1,⋯,αn)≥0 for all real α1,⋯,αn ''replaced by a suitable condition that reduces in the case n=1, X=Rn, to "f≥0''. This condition is as follows: For every analytic curve σ:(−ε,ε)→X for which (1) the germ of the image of σ at σ(0) is not contained in the germ of the singular set of X at σ(0) and (2) f∘σ(t)=atm+⋯(a≠0), one has a>0 and 2n∣m. The author studies the relationship between curves satisfying (1) and certain valuations on the field of meromorphic functions. On the other hand, Becker's theory of 2nth powers in formally real fields related the question of representing an element as the sum of 2nth powers to the study of valuations on the field. These two theories are combined to yield the 2nth power, analytic analogue of the original Hilbert problem.
  • Publication
    Algebraic versus analytic basicness
    (Walter de Gruyter, 1995) Andradas Heranz, Carlos; Ruiz Sancho, Jesús María; Broglia, Fabrizio; Galbiati, Margherita; Tognoli, Alberto
    A semialgebraic set is called basic if it can be described by a single system of strict polynomial inequalities. A semianalytic set is called basic if it can be described by a system of strict real analytic inequalities in a neighborhood of each of its points (the system, of course, may depend on the point). In the paper under review the authors describe a solution to the problem of geometric characterization of basic semialgebraic sets among the basic semianalytic sets. This solution appeared in their monograph [Mem. Amer. Math. Soc. 115 (1995), no. 553, vi+117 pp.]. The paper under review is not a summary of this monograph, but rather a complement to it. Instead of outlining the ideas of the proofs, the authors explain the notions of basic sets and their solution to the above problem with the help of many examples. The solution itself is formulated in terms of some collections of ultrafilters on the Boolean algebras of semialgebraic sets and of semianalytic germs; it is "geometric'' compared to the heavy algebraic machinery used in the proofs. The proofs themselves are not discussed in the paper, but the authors hope that their relatively elementary description of the results will motivate the reader to study the relevant algebra.
  • Publication
    Low dimensional sections of basic semialgebraic sets.
    (Univ Illinois Urbana-Champagn, 1994) Andradas Heranz, Carlos; Ruiz Sancho, Jesús María
    Let X be a real affine algebraic set and S a semialgebraic set. Many important results are known about the basicness of S: mainly, if S is basic open, S can be defined by s strict inequalities, where s is bounded by the dimension of X. It is also known (Br¨ocker-Scheiderer criterion) that an open semialgebraic set is basic and is defined by s inequalities (s-basic) if and only if for every irreducible subset Y of X the intersection S \ Y is generically s-basic. In a previous paper, the authors proved that to test the basicness of S it is sufficient to test the basicness of the intersections of S with every irreducible surface. In fact, this is the best possible result about the dimension, but it was conjectured by the authors that the first obstruction to s-basicness should be recognized in dimension s+1. In the present paper, the authors confirm this conjecture, and in fact they prove the following result: if S is basic, then S is s-basic if and only if for every irreducible subset Y ofX of dimension s+1 the intersection S \Y is generically s-basic. The proof of this theorem requires a deep analysis of real valuations. The results of independent interest about fans (algebroid fans) defined by means of power series and the approximation problem solved in this case are the main tools of the proof.
  • Publication
    Geometría proyectiva
    (Addison-Wesley Iberoamericana España, 1998) Rodríguez Sanjurjo, José Manuel; Ruiz Sancho, Jesús María
    Este libro presenta, de modo directo y accesible, las nociones y los resultados básicos de la geometría proyectiva: variedades y aplicaciones proyectivas, razón doble, homografías y cuádricas. La exposición teórica se completa con notas históricas aclaratorias y con una bibliografía exhaustiva. También se ofrece una extensa colección de más de cien problemas de índole variada. Es un libro especialmente indicado para estudiantes de la carrera de Matemáticas que deban abordar por primera vez estas materias.