The ring of bounded polynomials on a semi-algebraic set
HTML articles powered by AMS MathViewer
- by Daniel Plaumann and Claus Scheiderer PDF
- Trans. Amer. Math. Soc. 364 (2012), 4663-4682 Request permission
Abstract:
Let $V$ be a normal affine $\mathbb {R}$-variety, and let $S$ be a semi-algebraic subset of $V(\mathbb {R})$ which is Zariski dense in $V$. We study the subring $B_V (S)$ of $\mathbb {R}[V]$ consisting of the polynomials that are bounded on $S$. We introduce the notion of $S$-compatible completions of $V$, and we prove the existence of such completions when $\dim (V)\le 2$ or $S=V(\mathbb {R})$. An $S$-compatible completion $X$ of $V$ yields a ring isomorphism $\mathscr {O}_U(U)\overset {\sim }{\to } B_V(S)$ for some (concretely specified) open subvariety $U\supset V$ of $X$. We prove that $B_V(S)$ is a finitely generated $\mathbb {R}$-algebra if $\dim (V)\le 2$ and $S$ is open, and we show that this result becomes false in general when $\dim (V)\ge 3$.References
- Eberhard Becker, The real holomorphy ring and sums of $2n$th powers, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 139–181. MR 683132
- Eberhard Becker and Victoria Powers, Sums of powers in rings and the real holomorphy ring, J. Reine Angew. Math. 480 (1996), 71–103. MR 1420558, DOI 10.1515/crll.1996.480.71
- 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, DOI 10.1007/978-3-662-03718-8
- Steven Dale Cutkosky, Resolution of singularities, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004. MR 2058431, DOI 10.1090/gsm/063
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties. MR 637060
- Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
- S. Krug: Der Ring der globalen regulären Funktionen einer algebraischen Varietät. Diplomarbeit, Konstanz 2008.
- Shigeru Kuroda, A counterexample to the fourteenth problem of Hilbert in dimension three, Michigan Math. J. 53 (2005), no. 1, 123–132. MR 2125538, DOI 10.1307/mmj/1114021089
- M. Marshall, Approximating positive polynomials using sums of squares, Canad. Math. Bull. 46 (2003), no. 3, 400–418. MR 1994866, DOI 10.4153/CMB-2003-041-9
- Jean-Philippe Monnier, Anneaux d’holomorphie et Positivstellensatz archimédien, Manuscripta Math. 97 (1998), no. 3, 269–302 (French, with English summary). MR 1654768, DOI 10.1007/s002290050101
- Masayoshi Nagata, Addition and corrections to my paper “A treatise on the 14-th problem of Hilbert”, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 30 (1957), 197–200. MR 96645, DOI 10.1215/kjm/1250777056
- M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828
- D. Plaumann: Sums of squares on reducible real curves. Math. Z. (2010, to appear).
- Margherita Roggero, Sui sistemi lineari e il gruppo delle classi di divisori di una varietà reale, Ann. Mat. Pura Appl. (4) 135 (1983), 349–362 (1984) (Italian). MR 1553451, DOI 10.1007/BF01781076
- Claus Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics, vol. 1588, Springer-Verlag, Berlin, 1994. MR 1321819, DOI 10.1007/BFb0074269
- Claus Scheiderer, Sums of squares of regular functions on real algebraic varieties, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069. MR 1675230, DOI 10.1090/S0002-9947-99-02522-2
- Claus Scheiderer, Sums of squares on real algebraic curves, Math. Z. 245 (2003), no. 4, 725–760. MR 2020709, DOI 10.1007/s00209-003-0568-1
- Markus Schweighofer, Iterated rings of bounded elements and generalizations of Schmüdgen’s Positivstellensatz, J. Reine Angew. Math. 554 (2003), 19–45. MR 1952167, DOI 10.1515/crll.2003.004
- R. Vakil: An example of a nice variety whose ring of global sections is not finitely generated. Manuscript (2000), \urladdr{math.stanford.edu/ vakil/files/nonfg.pdf}
- O. Zariski, Interprétations algébrico-géométriques du quatorzième problème de Hilbert, Bull. Sci. Math. (2) 78 (1954), 155–168 (French). MR 65217
Additional Information
- Daniel Plaumann
- Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
- MR Author ID: 894950
- Email: daniel.plaumann@uni-konstanz.de
- Claus Scheiderer
- Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
- MR Author ID: 212893
- Email: claus.scheiderer@uni-konstanz.de
- Received by editor(s): February 9, 2010
- Received by editor(s) in revised form: July 29, 2010
- Published electronically: April 17, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 4663-4682
- MSC (2010): Primary 14P99; Secondary 14C20, 14E15, 14P05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05443-2
- MathSciNet review: 2922605