Evaluation codes defined by finite families of plane valuations at infinity

We construct evaluation codes given by weight functions defined over polynomial rings in m a parts per thousand yen 2 indeterminates. These weight functions are determined by sets of m-1 weight functions over polynomial rings in two indeterminates defined by plane valuations at infinity. Well-suited families in totally ordered commutative groups are an important tool in our procedureSupported by Spain Ministry of Education MTM2007-64704 and Bancaixa P1-1B2009-03. The authors thank to the referees for their valuable suggestions.Galindo Pastor, C.; Monserrat Delpalillo, FJ. (2014). Evaluation codes defined by finite families of plane valuations at infinity. Designs, Codes and Cryptography. 70(1-2):189-213. https://doi.org/10.1007/s10623-012-9738-7S189213701-2Abhyankar S.S.: Local uniformization on algebraic surfaces over ground field of characteristic p ≠ 0. Ann. Math. 63, 491–526 (1956)Abhyankar S.S.: On the valuations centered in a local domain. Am. J. 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Finite families of plane valuations: value semigroup, graded algebra and Poincaré series

[EN] In this paper, the authors are interested in some applications of valuation theory to algebraic geometry and, particularly, to singularity theory. The aim of this paper is to provide a concise survey of some aspects of the theory of plane valuations, adding some comments upon more general valuations when it is possible. For those valuations, authors describe value semigroup, graded algebra and Poincar ¿e series emphasizing on the recent study of the same algebraic objects for finite families of valuations and their relation with the corresponding ones for reduced germs of plane curvesSupported by Spain Ministry of Education MTM2007-64704 and Bancaixa P1-1B2009-03.Monserrat Delpalillo, FJ.; Galindo Pastor, C. (2012). Finite families of plane valuations: value semigroup, graded algebra and Poincaré series. Contemporary Mathematics. 566:189-212. https://doi.org/10.1090/conm/566/1122118921256

A class of polynomial planar vector fields with polynomial first integral

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm computes a minimal first integral. In addition, we solve the Poincaré problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field.The first author is partially supported by the MINECO/FEDER grant MTM2013-40998-P. The second and third authors are partially supported by the Spanish Ministry of Economy MTM2012-36917-C03-03 and Universitat Jaume I P1-1B2012-04 grants.Ferragut, A.; Galindo Pastor, C.; Monserrat Delpalillo, FJ. (2015). A class of polynomial planar vector fields with polynomial first integral. Journal of Mathematical Analysis and Applications. 430(1):354-380. https://doi.org/10.1016/j.jmaa.2015.04.062S354380430

Minimal plane valuations

[EN] We consider the value (mu) over cap(nu) = lim(m -> infinity) m(-1) a(mL), where a(mL) is the last value of the vanishing sequence of H-0(mL) along a divisorial or irrational valuation nu centered at O-P2,(p), L (respectively, p) being a line (respectively, a point) of the projective plane P-2 over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that (mu) over cap(nu) >= root 1/vol(nu) and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in [Comm. Anal. Geom. 25 (2017), pp. 125-161] to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original Nagata Conjecture. We also provide infinitely many families of minimal very general valuations with an arbitrary number of Puiseux exponents and an asymptotic result that can be considered as evidence in the direction of the above-mentioned conjecture.The authors were partially supported by the Spanish Government Ministerio de Economia, Industria y Competitividad/FEDER, grants MTM2012-36917-C03-03, MTM2015-65764-C3-2-P, and MTM2016-81735-REDT, as well as by Universitat Jaume I, grant P1-1B2015-02.Galindo Pastor, C.; Monserrat Delpalillo, FJ.; Moyano-Fernández, J. (2018). Minimal plane valuations. Journal of Algebraic Geometry. 27(4):751-783. https://doi.org/10.1090/jag/722S75178327

The Abhyankar-Moh theorem for plane valuations at infinity

We introduce the class of plane valuations at infinity and prove an analogue to the Abhyankar-Moh (semigroup) Theorem for it. © 2012 Elsevier Inc. All rights reserved.Supported by Spain Ministry of Education MTM2012-36917-C03-03 and Bancaixa P1-1B2009-03.Galindo Pastor, C.; Monserrat Delpalillo, FJ. (2013). The Abhyankar-Moh theorem for plane valuations at infinity. Journal of Algebra. 374:181-194. https://doi.org/10.1016/j.jalgebra.2012.11.001S18119437

The log-canonical threshold of a plane curve

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27].Supported by Spain Ministry of Economy MTM2012-36917-C03-03 and University Jaume I P1-1B2015-02.Galindo Pastor, C.; Hernando, F.; Monserrat Delpalillo, FJ. (2016). The log-canonical threshold of a plane curve. Mathematical Proceedings. 160(3):513-535. https://doi.org/10.1017/S0305004116000037S513535160

Crear y gestionar equipos deportivos: una experiencia innovadora de docencia y evaluación

El desarrollo de competencias profesionalizantes es uno de los principales objetivos del Espacio Europeo de Educación Superior, y ha supuesto un cambio sustancial no sólo en los Planes de Estudio de las titulaciones, sino también en los enfoques del proceso de enseñanza-aprendizaje. Así, el saber hacer y la didáctica de las competencias se convierten en el centro del esfuerzo docente mediante el empleo de metodologías como el Aprendizaje Basado en Problemas apoyadas en la gestión de una evaluación comprehensiva y participativa como es el empleo de rúbricas. Estas estrategias son el centro del esfuerzo docente para la enseñanza de competencias relacionadas con la creación y la gestión de los grupos en el contexto deportivo en la asignatura Psicología Social del Deporte del Grado en Ciencias del Deporte. Los resultados muestran que el proceso de Aprendizaje Basado en Problemas redunda en un entrenamiento eficaz de las competencias propias del manejo de los equipos deportivos. Futuras experiencias de innovacióndeberían seguir esta línea de trabajo, ampliando las competencias así entrenadas y combinando el ABP con otras estrategias, quizás, de carácter experiencial

Quasi-cyclic constructions of quantum codes

We give sufficient conditions for self-orthogonality with respect to symplec- tic, Euclidean and Hermitian inner products of a wide family of quasi-cyclic codes of index two. We provide lower bounds for the symplectic weight and the minimum distance of the involved codes. Supported in the previous results, we show algebraic constructions of good quantum codes and determine their parameter

Foliations with isolated singularities on Hirzebruch surfaces

[EN] We study foliations F on Hirzebruch surfaces Sd and prove that, similarly to those on the projective plane, any F can be represented by a bi-homogeneous polynomial affine 1-form. In case F has isolated singularities, we show that, for delta = 1, the singular scheme of F does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For delta not equal 1, we prove that the singular scheme of F does not determine the foliation. However, we prove that, in most cases, two foliations F and F' given by sections s and s' have the same singular scheme if and only if s' = Phi(s), for some global endomorphism F of the tangent bundle of S-delta.The first two authors are partially supported by the Spanish Government MICINN/FEDER/AEI/UE, grants PGC2018-096446-B-C22 and RED2018-102583-T, as well as by Generalitat Valenciana, grant AICO2019-223 and Universitat Jaume I, grantUJI-2018-10. The third authorwas partially supported by CONACYT: Estancias Sabaticas Vinculadas a la Consolidacion de Grupos de Investigacion, CVU 10069.Galindo Pastor, C.; Monserrat Delpalillo, FJ.; Olivares, J. (2021). Foliations with isolated singularities on Hirzebruch surfaces. Forum Mathematicum. 33(6):1471-1486. https://doi.org/10.1515/forum-2021-0135S1471148633