Integral closures in real algebraic geometry
Authors:
Goulwen Fichou, Jean-Philippe Monnier and Ronan Quarez
Journal:
J. Algebraic Geom. 30 (2021), 253-285
DOI:
https://doi.org/10.1090/jag/769
Published electronically:
July 31, 2020
MathSciNet review:
4233183
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Abstract |
References |
Additional Information
Abstract: We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety: the regular functions and the continuous rational functions.
References
- Carlos Andradas, Ludwig Bröcker, and Jesús M. Ruiz, Constructible sets in real geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 33, Springer-Verlag, Berlin, 1996. MR 1393194
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509
- Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
- Goulwen Fichou, Johannes Huisman, Frédéric Mangolte, and Jean-Philippe Monnier, Fonctions régulues, J. Reine Angew. Math. 718 (2016), 103–151 (French, with English and French summaries). MR 3545880, DOI https://doi.org/10.1515/crelle-2014-0034
- Goulwen Fichou, Jean-Philippe Monnier, and Ronan Quarez, Continuous functions on the plane regular after one blowing-up, Math. Z. 285 (2017), no. 1-2, 287–323. MR 3598813, DOI https://doi.org/10.1007/s00209-016-1708-8
- G. Fichou, J.-P. Monnier, R. Quarez, Weak and semi normalization in real algebraic geometry, arXiv:1706.04467, (2018)
- János Kollár, Wojciech Kucharz, and Krzysztof Kurdyka, Curve-rational functions, Math. Ann. 370 (2018), no. 1-2, 39–69. MR 3747483, DOI https://doi.org/10.1007/s00208-016-1513-z
- János Kollár and Krzysztof Nowak, Continuous rational functions on real and $p$-adic varieties, Math. Z. 279 (2015), no. 1-2, 85–97. MR 3299844, DOI https://doi.org/10.1007/s00209-014-1358-7
- Wojciech Kucharz, Rational maps in real algebraic geometry, Adv. Geom. 9 (2009), no. 4, 517–539. MR 2574137, DOI https://doi.org/10.1515/ADVGEOM.2009.024
- Wojciech Kucharz and Krzysztof Kurdyka, Stratified-algebraic vector bundles, J. Reine Angew. Math. 745 (2018), 105–154. MR 3881474, DOI https://doi.org/10.1515/crelle-2015-0105
- Wojciech Kucharz and Krzysztof Kurdyka, From continuous rational to regulous functions, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 719–747. MR 3966787
- Krzysztof Kurdyka, Ensembles semi-algébriques symétriques par arcs, Math. Ann. 282 (1988), no. 3, 445–462 (French). MR 967023, DOI https://doi.org/10.1007/BF01460044
- Krzysztof Kurdyka and Adam Parusiński, Arc-symmetric sets and arc-analytic mappings, Arc spaces and additive invariants in real algebraic and analytic geometry, Panor. Synthèses, vol. 24, Soc. Math. France, Paris, 2007, pp. 33–67 (English, with English and French summaries). MR 2409688
- Frédéric Mangolte, Variétés algébriques réelles, Cours Spécialisés [Specialized Courses], vol. 24, Société Mathématique de France, Paris, 2017 (French). MR 3727103
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- Jean-Philippe Monnier, Semi-algebraic geometry with rational continuous functions, Math. Ann. 372 (2018), no. 3-4, 1041–1080. MR 3880292, DOI https://doi.org/10.1007/s00208-018-1679-7
- I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR 0366917
References
- Carlos Andradas, Ludwig Bröcker, and Jesús M. Ruiz, Constructible sets in real geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 33, Springer-Verlag, Berlin, 1996. MR 1393194
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, translated from the 1987 French original, revised by the authors, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. MR 1659509
- Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348
- David Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960
- Goulwen Fichou, Johannes Huisman, Frédéric Mangolte, and Jean-Philippe Monnier, Fonctions régulues, J. Reine Angew. Math. 718 (2016), 103–151 (French, with English and French summaries). MR 3545880, DOI https://doi.org/10.1515/crelle-2014-0034
- Goulwen Fichou, Jean-Philippe Monnier, and Ronan Quarez, Continuous functions on the plane regular after one blowing-up, Math. Z. 285 (2017), no. 1-2, 287–323. MR 3598813, DOI https://doi.org/10.1007/s00209-016-1708-8
- G. Fichou, J.-P. Monnier, R. Quarez, Weak and semi normalization in real algebraic geometry, arXiv:1706.04467, (2018)
- János Kollár, Wojciech Kucharz, and Krzysztof Kurdyka, Curve-rational functions, Math. Ann. 370 (2018), no. 1-2, 39–69. MR 3747483, DOI https://doi.org/10.1007/s00208-016-1513-z
- János Kollár and Krzysztof Nowak, Continuous rational functions on real and $p$-adic varieties, Math. Z. 279 (2015), no. 1-2, 85–97. MR 3299844, DOI https://doi.org/10.1007/s00209-014-1358-7
- Wojciech Kucharz, Rational maps in real algebraic geometry, Adv. Geom. 9 (2009), no. 4, 517–539. MR 2574137, DOI https://doi.org/10.1515/ADVGEOM.2009.024
- Wojciech Kucharz and Krzysztof Kurdyka, Stratified-algebraic vector bundles, J. Reine Angew. Math. 745 (2018), 105–154. MR 3881474, DOI https://doi.org/10.1515/crelle-2015-0105
- Wojciech Kucharz and Krzysztof Kurdyka, From continuous rational to regulous functions, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 719–747. MR 3966787
- Krzysztof Kurdyka, Ensembles semi-algébriques symétriques par arcs, Math. Ann. 282 (1988), no. 3, 445–462 (French). MR 967023, DOI https://doi.org/10.1007/BF01460044
- Krzysztof Kurdyka and Adam Parusiński, Arc-symmetric sets and arc-analytic mappings, Arc spaces and additive invariants in real algebraic and analytic geometry, Panor. Synthèses, vol. 24, Soc. Math. France, Paris, 2007, pp. 33–67 (English, with English and French summaries). MR 2409688
- Frédéric Mangolte, Variétés algébriques réelles, Cours Spécialisés [Specialized Courses], vol. 24, Société Mathématique de France, Paris, 2017 (French). MR 3727103
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- Jean-Philippe Monnier, Semi-algebraic geometry with rational continuous functions, Math. Ann. 372 (2018), no. 3-4, 1041–1080. MR 3880292, DOI https://doi.org/10.1007/s00208-018-1679-7
- I. R. Shafarevich, Basic algebraic geometry, translated from the Russian by K. A. Hirsch, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. MR 0366917
Additional Information
Goulwen Fichou
Affiliation:
Université de Rennes, IRMAR (UMR 6625), Campus de Beaulieu, 35042 Rennes Cedex, France
MR Author ID:
679660
Email:
goulwen.fichou@univ-rennes1.fr
Jean-Philippe Monnier
Affiliation:
LUNAM Université, LAREMA, Université d’Angers
MR Author ID:
622950
ORCID:
0000-0002-4810-6686
Email:
jean-philippe.monnier@univ-angers.fr
Ronan Quarez
Affiliation:
Université de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France
MR Author ID:
630274
Email:
ronan.quarez@univ-rennes1.fr
Received by editor(s):
December 19, 2018
Received by editor(s) in revised form:
March 9, 2020, and March 12, 2020
Published electronically:
July 31, 2020
Additional Notes:
The authors benefitted from the support of the Centre Henri Lebesgue ANR-11-LABX-0020-01, and the first two authors from the project ENUMGEOM ANR-18-CE40-0009.
Article copyright:
© Copyright 2020
University Press, Inc.
