×

Metrizability of the space of \(\mathbb{R}\)-places of a real function field. (English) Zbl 1201.12001

The topological space \(M_F\) of \(\mathbb R\)-places of the field \(F\) is a quotient space of the space \(X_F\) of all orderings of \(F.\) In general its structure can be very complicated. For \(F={\mathbb R}(X)\) the space \(M_F\) is the real projective line. However for \(n>1\) the structure of \(M_{{\mathbb R}(X_1,\dots,X_n)}\) is not known so far. The main theorem of the paper says that for any uncountable real closed field \(R\), the space of \(\mathbb R\)-real places of the rational function field \(R(X_1,X_2)\) is not metrizable. The authors also consider the more general question of when the space of \(\mathbb R\)-places of a finitely generated formally real field extension \(F\) of a real closed field \(R\) is metrizable. They prove that if \(R\) is uncountable and the transcendence degree of \(F\) over \(R\) is greater than 1, then \(M_F\) is not metrizable, whereas in the case of transcendence 1 and \(R\) being Archimedean the space \(M_F\) is metrizable if \(R\) is countable.

MSC:

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
14P05 Real algebraic sets
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alonso M.E., Gamboa J.M., Ruiz J.M.: On orderings in real surfaces. J. Pure Appl. Algebra 36, 1–14 (1985) · Zbl 0559.14015 · doi:10.1016/0022-4049(85)90059-3
[2] Becker, E.: Valuations and real places in the theory of formally real fields. In: Géométrie Algébrique Réelle et Formes Quadratiques, Proceedings of Conference in Rennes. Lecture Notes in Mathematics, vol. 959, pp. 1–40. Springer, Berlin (1982)
[3] Becker, E., Gondard, D.: Notes on the space of real places of a formally real field. In: Real Analytic and Algebraic Geometry, Trento Conference, 1992, pp. 21–46. Walter de Gruyter, Berlin (1995) · Zbl 0869.12002
[4] Brown R.: Real-valued places on the function field of an algebraic curve. Houst. J. Math. 6, 227–243 (1980) · Zbl 0455.14016
[5] Craven T.: The topological space of orderings of a rational function field. Duke Math. J. 41, 339–347 (1974) · Zbl 0293.12010 · doi:10.1215/S0012-7094-74-04139-8
[6] Gilmer R.: Extension of an order to a simple transcendental extension. Contemp. Math. 8, 113–118 (1982) · Zbl 0483.12006
[7] Kelley J.: General Topology. Graduate Texts in Mathematics. Springer, Berlin (1955) · Zbl 0066.16604
[8] Knebusch, M., Scheiderer, C.: Einführung in die reelle algebra. Vieweg Studium Aufbaukurs Mathematik 63 (1989) · Zbl 0732.12001
[9] Kuhlmann F.-V., Kuhlmann S., Marshall M., Zekavat M.: Embedding ordered fields in formal power series fields. J. Pure Appl. Algebra 169, 71–90 (2002) · Zbl 0998.12010 · doi:10.1016/S0022-4049(01)00064-0
[10] Kuhlmann, F.-V., Machura, M., Osiak, K.: Spaces of \({\mathbb{R}}\) -places of function fields over real closed fields. Preprint (2009) · Zbl 1263.12001
[11] Lam, T.-Y.: Orderings, valuations and quadratic forms. In: CBMS Regional Conference Series in Mathematics, vol. 52. AMS, Providence (1983) · Zbl 0516.12001
[12] Machura, M., Osiak, K.: Spaces of \({\mathbb{R}}\) -places of rational function fields. arXiv:0803.0676 (2008) · Zbl 1223.12002
[13] Marshall, M.: Spaces of Orderings and Abstract Real Spectra. Lecture Notes in Mathematics, vol. 1636. Springer, Berlin (1996) · Zbl 0866.12001
[14] Marshall, M.: Positive Polynomials and Sums of Squares. Surveys and Monographs, vol. 146. AMS, Providence (2008) · Zbl 1169.13001
[15] Schülting, H.-W.: Real holomorphy rings in real algebraic geometry. In: Géométrie Algébrique Réelle et Formes Quadratiques, Proceedings of Conference in Rennes. Lecture Notes in Mathematics, vol. 959, pp. 433–442. Springer, Berlin (1982)
[16] Zekavat, M.: Orderings, cuts and formal power series. Doctoral Thesis, University of Saskatchewan (2000). http://library2.usask.ca/theses/
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.