### Browsing by Author "Gamboa, J. M."

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Publication A Characterization Of Rational And Elliptic Real Algebraic-Curves In Terms Of Their Space Of Orderings(Rocky Mountain Journal Of Mathematics, 1984) Gamboa, J. M.Let K be a formally real field with space of orderings X(K). Then Aut(K) operates on X(K). K is said to have the ”dense orbits property” if for any x 2 X(K) the orbit of x is dense in X(K). Fields with the dense orbits property were introduced by D. W. Dubois and T. Recio in Contemp. Math. 8, 265-288 (1982; Zbl 0484.12003) under the name of Q1-fields. They were further studied by the author and T. Recio [J. Pure Appl. Algebra 30, 237-246 (1983; Zbl 0533.12018)]. In the present paper the dense orbits property is studied for function fields of real algebraic varieties. So, let V be a real algebraic variety over the field R of real numbers, R(V ) the function field of V. It is proved that Aut(R(V )) is infinite if R(V ) has the dense orbits property. If V is a curve then R(V ) has the dense orbits property if and only if V is a rational or elliptic curve.Publication A Note On Cohomology Over Non Algebraically Closed Fields(Australian Mathematics, 1999) Gamboa, J. M.We characterise algebraically closed fields as those for which the first cohomology group tf^fc^On) of the sheaf On of regular functions over kn vanishes for all positive intergers n.Publication A note on hyperplane sections of real algebraic sets(Sociedad Matemática Mexicana, 1984) Gamboa, J. M.The author studies the size of the set of hyperplanes which meet a non- zero-dimensional algebraic set V over a real-closed ground field R. More precisely, let us denote by $V\sb c$ the locus of central points of V, i.e., the closure, in the order topology of $R\sp n$, of the set of regular points of V. The author proves the following: There exists a linear isomorphism $\sigma$ of $R\sp n$ such that for every ``generic'' hyperplane H of $R\sp n$, either H meets $V\sb c$ or its transform by $\sigma$ meets $V\sb c$.Publication A note on projections of real algebraic varieties.(Pacific Journal of Mathematics, 1984) Andradas Heranz, Carlos; Gamboa, J. M.We prove that any regularly closed semialgebraic set of R", where R is any real closed field and regularly closed means that it is the closure of its interior, is the projection under a finite map of an irreducible algebraic variety in some Rn + k. We apply this result to show that any clopen subset of the space of orders of the field of rational functions K= R(X1,...iXn) is the image of the space of orders of a finite extension of K.Publication A Positivstellensatz For Rings Of Continuous-Functions(Elsevier Science, 1987) Gamboa, J. M.A one page proof (using the real spectrum) of the following result: Let f, g : X ! R be a continuous functions on a topological space X, such that f is strictly positive on g−1(0). Then there exist continuous functions u, v and w such that (1 + v2)f = 1 + w2 + gu.Publication Algebra lineal (2ª ed.)(Sanz y Torres, 2010) Fernando Galván, José Francisco; Gamboa, J. M.; Ruiz Sancho, Jesús MaríaPublication Algebraic function fields in one variable over real-closed group fields(Hermann, 1987) Bujalance, E.; Etayo Gordejuela, J. Javier; Gamboa, J. M.; Aroca, J.M.; Sánchez-Giralda, T.; Vincente, J.L.Publication An Algorithm To Compute Odd Orders And Ramification Indexes Of Cyclic Actions On Compact Surfaces(Springer, 1994) Gamboa, J. M.; Bujalance, E.; Costa Gonzalez, A.F.; Lafuente López, JavierIn this paper we get an effective algorithm to compute all odd orders and ramification indices of homeomorphismsPublication An algorithm to compute orders and ramification indices of cyclic actions on compact surfaces .2.(Springer Verlag, 1996) Gamboa, J. M.; Bujalance, E.; Costa, F.A.; Lafuente López, JavierIn this paper we obtain an effective algorithm to compute all even orders and ramification indices of homeomorphisms of finite order acting on compact surfaces, orientable or not. This completes the case of odd orders, previously studied by the authors.Publication An Example Of A Constructible Set With Non Constructible Closure(Elsevier, 1988) Gamboa, J. M.Publication Automorphism groups of compact bordered Klein surfaces. A combinatorial approach(Springer, 1990) Bujalance, E.; Etayo Gordejuela, J. Javier; Gamboa, J. M.; Gromadzki, G.Publication Automorphism groups of Klein surfaces without involutions(Universidad Complutense de Madrid, 1986) Bujalance, E.; Etayo Gordejuela, J. Javier; Gamboa, J. M.; Outerelo Dominguez, EnriqueThe authors describe in terms of non-Euclidean crystallographic groups all Klein surfaces whose automorphism group is one of the following: Z/p⊕⋯⊕Z/p , Z/pq , or Z/p 2 , where p and q are distinct odd primes. This includes every nontrivial finite group of order less than 21, so they are able to use their results to find all topological types of Klein surfaces of algebraic genus less than 22 whose automorphism group has odd order bigger than one. This list takes 29 pages! They note that the cyclic groups of orders 13, 17 and 19 do not appear, a result of some interest as these groups certainly act as a subgroup of the automorphism group of a surface of algebraic genus less than 22.Publication Automorphism Groups Of The Real Projective Plane With Holes And Their Conjugacy Classes Within Its Mapping Class Group(Springer, 2005) Gamboa, J. M.; Bujalance, E.; Cirre, F.J.For each integer g ≥ 2 we give the complete list of groups acting as a group of dianalytic automorphisms of a real projective plane with g holes, which, in algebraic terms, correspond to birational automorphisms of real algebraic (M − 1)-curves. We also determine those acting as the full group of automorphisms of such a surface. Furthermore, the conjugacy classes of the finite subgroups of its mapping class group are calculated.Publication Automorphism-Groups Of Algebraic-Curves Of Rn Of Genus-2(Birkhauser Verlag, 1984) Gamboa, J. M.; Bujalance, E.Publication Automorphism-Groups Of Real Algebraic-Curves Of Genus-3(Japan Acad, 1986) Gamboa, J. M.; Bujalance, E.; Etayo Gordejuela, J. JavierLet C be an algebraic curve of genus 3, defined over the real field R. The automorphism group of C is studied in this paper. In a paper by the same authors [Mich. Math. J. 33, 55-74 (1986; see 20043 below)], the hyperelliptic case was solved, the authors found 5 abstract groups in that case. In the paper under review, a full classification is announced, in which 8 abstract groups appear. A sketch of one of the new cases is given, full proofs appeared in Mem. R. Acad. Cienc. Exactas Fis. Nat. Madrid, Ser. Cienc. Exactas 19 (1985; see the following review)].Publication Automorphisms of Klein surfaces with fixed points.(Cambridge, 1999) Corrales Rodrigáñez, Carmen; Gamboa, J. M.; Gromadzki, G.Publication Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves(Real Academia de Ciencias Exactas, Físicas y Naturales, 1991) Gamboa, J. M.This paper gives a very good systematic presentation of the equivalence between the algebraic function fields in one variable over the field $\bbfR$ of real numbers and the Klein surfaces. In section 1 Klein surfaces and morphisms between them are defined, and example as well as the basic facts about them are given. The double covering of a Klein surface and the quotient of a Riemann surface under an antianalytic involution is described, and it is noted that these two constructions are mutually inverse. Section 2 is devoted to the notion of a meromorphic function of a compact Klein surface. It is shown that the field of meromorphic functions of a compact Klein surface is an algebraic function field in one variable over $\bbfR$. Also there exists a functor of the category $\cal K$ of compact Klein surfaces to the category ${\cal F}\sb \bbfR$ of the algebraic function fields in one variable over $\bbfR$. An intensive study of the set $S(E\mid\bbfR)$ of proper valuation rings $V$ of $E\in{\cal F}\sb \bbfR$ with $V\supset\bbfR$ is the object of section 3. The main results of this section are:\par (a) The residue field of $V\in S(E\mid\bbfR)$ is $\bbfR$ iff $E$ admits some ordering with respect to which $V$ is convex.\par (b) The Riemann theorem about the dimension $\ell(L)$ of the space $L(D)$ associated to the divisor $D$ of $E\mid \bbfR$.\par With these notations it is proved in section 4 that $S(E\mid\bbfR)$ admits a unique structure of a Klein surface for which $p:S(E(\sqrt{- 1})(\bbfC)\to S(E\mid\bbfR)$ is a morphism of Klein surfaces and $M(S(E\mid\bbfR))=E$. Further it is shown that every compact Klein surface $S$ is isomorphic to $S(M(S)\mid\bbfR)$. Also: $S\mapsto M(S)$ and $E\mapsto S(E\mid\bbfR)$ give an equivalence between $\cal K$ and ${\cal F}\sb \bbfR$. Here the Klein surfaces with non empty boundary correspond to the formally real fields. Among a series of interesting comments and remarks we mention merely two:\par (i) There are several non homeomorphic curves with the same field of rational functions, but there is a unique one among them which is irreducible, compact, non-singular and affine.\par (ii) The Klein surface $S$ with empty boundary is orientable iff $M(S)$ contains $\bbfC$.\par This carefully written paper is very interesting and recommended even for specialists.Publication Conquest of the plane : using the economics pack "Applications of Mathematica" for a didactic primer on analytic geometry and calculus [book review](European Mathematical Society, 2011) Gamboa, J. M.Publication Curvas algebraicas reales vistas como superficies de Klein compactas con borde: un ejemplo(Universidad Complutense de Madrid, 1994) Gamboa, J. M.Publication Curvas algebraicas reales y superficies de Klein(Real Academia de Ciencias Exactas, Físicas y Naturales, 1984) Bujalance, E.; Etayo Gordejuela, J. Javier; Gamboa, J. M.The classical correspondence between Riemann surfaces and complex algebraic curves, extends by the work of Ailing and Greenleaf to Klein surfaces and real algebraic curves. The topological invariants of the surface determine the ones of a smonoth and bounded model of the associated curve, and conversely. Moreover, the fields of meromorphic functions of both coincide. So, the automorphisms group, the real part of the associated complex curve, and the coverings and moduli space of the curve, may be studied in terms of the automorphisms group, the symmetries, the coverings and the Teichmüller space of the associated surface.