Abstract
We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bilski, M., Kucharz, W., Valette, A., Valette, G.: Vector bundles and reguluous maps. Math. Z. 275, 403–418 (2013)
Bochnak, J., Coste M., Roy M.-F.: Real Algebraic Geometry, Ergeb. der Math. und ihrer Grenzgeb, Folge 3, vol. 36. Springer, Berlin (1998)
Brenner, H.: Continuous solutions to algebraic forcing equations. 2006. arXiv:math/0608611 [math.AC]
Epstein, N., Hochster, M.: Continuous closure, axes closure, and natural closure. 2011. arXiv:1106.3462v2 [math.AC]
Fefferman, C., Kollár, J.: Continuous solutions of linear equations. In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry, pp. 233–282. Dev. Math. 28, Springer, New York (2013)
Fichou, G., Huismann, J., Mangolte, F., Monnier, J.-P.: Fonctions régulues. J. Reine Angew. Math. 718, 103–151 (2016)
Fichou, G., Monnier, J.-P., Quarez, R.: Continuous functions in the plane regular after one blowing-up. Math. Z. 285, 287–323 (2017)
Iitaka, S.: Algebraic Geometry. An Introduction to Birational Geometry of Algebraic Varieties. Springer, Berlin (1982)
Kollár, J.: Continuous closure of sheaves. Mich. Math. J. 61, 475–491 (2012)
Kollár, J., Kucharz, W., Kurdyka, K.: Curve-rational functions. Math. Ann. (2017). doi:10.1007/s00208-016-1513-z
Kollár, J., Nowak, K.: Continuous rational functions on real and \(p\)-adic varieties. Math. Z. 279, 85–97 (2015)
Kucharz, W.: Rational maps in real algebraic geometry. Adv. Geom. 9, 517–539 (2009)
Kucharz, W.: Regular versus continuous rational maps. Topol. Appl. 160, 1375–1378 (2013)
Kucharz, W.: Approximation by continuous rational maps into spheres. J. Eur. Math. Soc. 16, 1555–1569 (2014)
Kucharz, W.: Continuous rational maps into the unit \(2\)-sphere. Arch. Math. (Basel) 102, 257–261 (2014)
Kucharz, W.: Continuous rational maps into spheres. Math. Z. 283, 1201–1215 (2016)
Kucharz, W., Kurdyka, K.: Stratified-algebraic vector bundles. J. Reine Angew. Math. (2016), doi:10.1515/crelle-2015-0105, arXiv:1308.4376 [math.AG]
Kucharz, W., Kurdyka, K.: Comparison of stratified-algebraic and topological K-theory. 2015. arXiv:1511.04238 [math.AG]
Nowak, K.J.: Some results of algebraic geometry over Henselian rank one valued fields. Sel. Math. New Ser. 28, 455–495 (2017). doi:10.1007/s00029-016-0245-y
Zieliński, M.: Homotopy properties of some real algebraic maps. Homol. Homotopy Appl. 18, 287–299 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Wojciech Kucharz was partially supported by the Natinal Science Centre (Poland), under grant number 2014/15/B/ST1/00046. He also acknowledges with gratitude support and hospitality of the Max–Planck–Institut für Mathematik in Bonn.
Krzysztof Kurdyka was partially supported by ANR project STAAVF (France).
Rights and permissions
About this article
Cite this article
Kucharz, W., Kurdyka, K. Linear equations on real algebraic surfaces. manuscripta math. 154, 285–296 (2017). https://doi.org/10.1007/s00229-017-0925-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-017-0925-8