Sums of squares in real analytic rings
HTML articles powered by AMS MathViewer
- by José F. Fernando PDF
- Trans. Amer. Math. Soc. 354 (2002), 1909-1919 Request permission
Abstract:
Let $A$ be an analytic ring. We show: (1) $A$ has finite Pythagoras number if and only if its real dimension is $\leq 2$, and (2) if every positive semidefinite element of $A$ is a sum of squares, then $A$ is real and has real dimension $2$.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, DOI 10.1007/978-3-642-80024-5
- Carlos Andradas and Jesús M. Ruiz, On local uniformization of orderings, Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991) Contemp. Math., vol. 155, Amer. Math. Soc., Providence, RI, 1994, pp. 19–46. MR 1260700, DOI 10.1090/conm/155/01372
- 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
- Antonio Campillo and Jesús M. Ruiz, Some remarks on Pythagorean real curve germs, J. Algebra 128 (1990), no. 2, 271–275. MR 1036389, DOI 10.1016/0021-8693(90)90021-F
- M. D. Choi, Z. D. Dai, T. Y. Lam, and B. Reznick, The Pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math. 336 (1982), 45–82. MR 671321, DOI 10.1515/crll.1982.336.45
- J.F. Fernando: Positive semidefinite germs in real analytic surfaces, Math. Ann. (to appear).
- J.F. Fernando: On the Pythagoras numbers of real analytic rings, J. Algebra 243, 321-338 (2001).
- J.F. Fernando: Analytic surface germs with minimal Pythagoras number, in preparation.
- J.F. Fernando, J.M. Ruiz: Positive semidefinite germs on the cone, Pacific J. Math. (to appear).
- Theo de Jong and Gerhard Pfister, Local analytic geometry, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 2000. Basic theory and applications. MR 1760953, DOI 10.1007/978-3-322-90159-0
- Jesus J. Ortega, On the Pythagoras number of a real irreducible algebroid curve, Math. Ann. 289 (1991), no. 1, 111–123. MR 1087240, DOI 10.1007/BF01446562
- R. Quarez: Pythagoras numbers of real algebroid curves and Gram matrices. J. Algebra 238, 139-158 (2001).
- Jesús M. Ruiz, On Hilbert’s 17th problem and real Nullstellensatz for global analytic functions, Math. Z. 190 (1985), no. 3, 447–454. MR 806902, DOI 10.1007/BF01215144
- Jesús M. Ruiz, The basic theory of power series, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1993. MR 1234937, DOI 10.1007/978-3-322-84994-6
- Jesús M. Ruiz, Sums of two squares in analytic rings, Math. Z. 230 (1999), no. 2, 317–328. MR 1676722, DOI 10.1007/PL00004692
- 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
- C. Scheiderer: On sums of squares in local rings, J. Reine Angew. Math. 540, 205–227 (2001).
Additional Information
- José F. Fernando
- Affiliation: Departamento Algebra, Facultad Ciencias Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain
- Email: josefer@mat.ucm.es
- Received by editor(s): March 29, 2001
- Received by editor(s) in revised form: August 14, 2001
- Published electronically: January 10, 2002
- Additional Notes: Research partially supported by DGICYT, PB98-0756-C02-01
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 1909-1919
- MSC (2000): Primary 11E25; Secondary 14P15
- DOI: https://doi.org/10.1090/S0002-9947-02-02956-2
- MathSciNet review: 1881023