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.
@article{bwmeta1.element.doi-10_2478_s11533-014-0409-y,
author = {Taras Banakh and Yaroslav Kholyavka and Oles Potyatynyk and Micha\l\ Machura and Katarzyna Kuhlmann},
title = {On the dimension of the space of $\mathbb{R}$-places of certain rational function fields},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1239-1248},
zbl = {1311.12004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0409-y}
}
Taras Banakh; Yaroslav Kholyavka; Oles Potyatynyk; Michał Machura; Katarzyna Kuhlmann. On the dimension of the space of ℝ-places of certain rational function fields. Open Mathematics, Tome 12 (2014) pp. 1239-1248. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0409-y/
[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.2012.09.012 | Zbl 1275.54022
[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 | Zbl 0869.12002
[3] Bochnak J., Coste M., Roy M.-F., Real Algebraic Geometry, Ergeb. Math. Grenzgeb., 36, Springer, Berlin, 1998 http://dx.doi.org/10.1007/978-3-662-03718-8
[4] Brown R., Real places and ordered fields, Rocky Mountain J. Math., 1971, 1(4), 633–636 http://dx.doi.org/10.1216/RMJ-1971-1-4-633
[5] Coste M., Real algebraic sets, available at http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf | Zbl 1144.14305
[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-X | Zbl 0315.12106
[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
[8] Dranishnikov A., Dydak J., Extension dimension and extension types, Proc. Steklov Inst. Math., 1996, 1(212), 55–88 | Zbl 0874.54033
[9] Dubois D.W., Infinite Primes and Ordered Fields, Dissertationes Math. Rozprawy Mat., 69, Polish Academy Sciences, Warsaw, 1970 | Zbl 0266.12103
[10] Engelking R., Theory of Dimensions Finite and Infinite, Sigma Ser. Pure Math., 10, Heldermann, Lemgo, 1995 | Zbl 0872.54002
[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 | Zbl 1262.12002
[12] Lam T.Y., Orderings, Valuations and Quadratic Forms, CBMS Regional Conf. Ser. in Math., 52, American Mathematical Society, Providence, 1983
[13] Lang S., The theory of real places, Ann. of Math., 1953, 57(2), 378–391 http://dx.doi.org/10.2307/1969865 | Zbl 0052.03301
[14] Lang S., Algebra, Grad. Text in Math., 221, Springer, New York, 2002 http://dx.doi.org/10.1007/978-1-4613-0041-0
[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-z | Zbl 1201.12001
[16] Marshall M., Positive Polynomials and Sums of Squares, Math. Surveys Monogr., 146, American Mathematical Society, Providence, 2008
[17] Schleiermacher A., On Archimedean fields, J. Geom., 2009, 92(1–2), 143–173 http://dx.doi.org/10.1007/s00022-008-2098-9 | Zbl 1201.12004