Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 12, Issue 8

Issues

On the dimension of the space of ℝ-places of certain rational function fields

Taras Banakh
  • Department of Mathematics, Ivan Franko National University of Lviv, 1 Universytetska St., Lviv, 79000, Ukraine
  • Instytut Matematyki, Jan Kochanowski University, ul. Świętokrzyska 15, 25-406, Kielce, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yaroslav Kholyavka / Oles Potyatynyk / Michał Machura / Katarzyna Kuhlmann
Published Online: 2014-05-08 | DOI: https://doi.org/10.2478/s11533-014-0409-y

Abstract

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.

MSC: 12F20; 12J15; 54F45; 55M10

Keywords: Space of R-places; Graphoid; Dimension; Cohomological dimension; Extension dimension

  • [1] Banakh T., Potyatynyk O., Dimension of graphoids of rational vector-functions, Topology Appl., 2013, 160(1), 24–44 http://dx.doi.org/10.1016/j.topol.2012.09.012CrossrefGoogle Scholar

  • [2] Becker E., Gondard D., Notes on the space of real places of a formally real field, In: Real Analytic and Algebraic Geometry, Trento, September 21–25, 1992, de Gruyter, Berlin, 1995, 21–46 Google Scholar

  • [3] Bochnak J., Coste M., Roy M.-F., Real Algebraic Geometry, Ergeb. Math. Grenzgeb., 36, Springer, Berlin, 1998 http://dx.doi.org/10.1007/978-3-662-03718-8CrossrefGoogle Scholar

  • [4] Brown R., Real places and ordered fields, Rocky Mountain J. Math., 1971, 1(4), 633–636 http://dx.doi.org/10.1216/RMJ-1971-1-4-633CrossrefGoogle Scholar

  • [5] Coste M., Real algebraic sets, available at http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf Google Scholar

  • [6] Craven T.C., The Boolean space of orderings of a field, Trans. Amer. Math. Soc., 1975, 209, 225–235 http://dx.doi.org/10.1090/S0002-9947-1975-0379448-XCrossrefGoogle Scholar

  • [7] Dranishnikov A.N., Cohomological dimension theory of compact metric spaces, Topology Atlas Invited Contributions, 2001, 6(3), 7–73, available at http://at.yorku.ca/t/a/i/c/43.htm Google Scholar

  • [8] Dranishnikov A., Dydak J., Extension dimension and extension types, Proc. Steklov Inst. Math., 1996, 1(212), 55–88 Google Scholar

  • [9] Dubois D.W., Infinite Primes and Ordered Fields, Dissertationes Math. Rozprawy Mat., 69, Polish Academy Sciences, Warsaw, 1970 Google Scholar

  • [10] Engelking R., Theory of Dimensions Finite and Infinite, Sigma Ser. Pure Math., 10, Heldermann, Lemgo, 1995 Google Scholar

  • [11] Kuhlmann K., The structure of spaces of R-places of rational function fields over real closed fields, preprint available at http://math.usask.ca/fvk/recpap.htm Google Scholar

  • [12] Lam T.Y., Orderings, Valuations and Quadratic Forms, CBMS Regional Conf. Ser. in Math., 52, American Mathematical Society, Providence, 1983 Google Scholar

  • [13] Lang S., The theory of real places, Ann. of Math., 1953, 57(2), 378–391 http://dx.doi.org/10.2307/1969865CrossrefGoogle Scholar

  • [14] Lang S., Algebra, Grad. Text in Math., 221, Springer, New York, 2002 http://dx.doi.org/10.1007/978-1-4613-0041-0CrossrefGoogle Scholar

  • [15] Machura M., Marshall M., Osiak K., Metrizability of the space of R-places of a real function field, Math. Z., 2010, 266(1), 237–242 http://dx.doi.org/10.1007/s00209-009-0566-zWeb of ScienceCrossrefGoogle Scholar

  • [16] Marshall M., Positive Polynomials and Sums of Squares, Math. Surveys Monogr., 146, American Mathematical Society, Providence, 2008 Google Scholar

  • [17] Schleiermacher A., On Archimedean fields, J. Geom., 2009, 92(1–2), 143–173 http://dx.doi.org/10.1007/s00022-008-2098-9CrossrefGoogle Scholar

About the article

Published Online: 2014-05-08

Published in Print: 2014-08-01


Citation Information: Open Mathematics, Volume 12, Issue 8, Pages 1239–1248, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-014-0409-y.

Export Citation

© 2014 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in