On the finiteness of Pythagoras numbers of real meromorphic functions

Francesca Acquistapace; Fabrizio Broglia; José F. Fernando; Jesús M. Ruiz

Bulletin de la Société Mathématique de France (2010)

  • Volume: 138, Issue: 2, page 231-247
  • ISSN: 0037-9484

Abstract

top
We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real meromorphic power series. This measures the difficulty of the 17th Hilbert problem in the analytic case.

How to cite

top

Acquistapace, Francesca, et al. "On the finiteness of Pythagoras numbers of real meromorphic functions." Bulletin de la Société Mathématique de France 138.2 (2010): 231-247. <http://eudml.org/doc/272513>.

@article{Acquistapace2010,
abstract = {We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real meromorphic power series. This measures the difficulty of the 17th Hilbert problem in the analytic case.},
author = {Acquistapace, Francesca, Broglia, Fabrizio, Fernando, José F., Ruiz, Jesús M.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {17th Hilbert problem; pythagoras number; sum of squares; bad set; germs at closed sets},
language = {eng},
number = {2},
pages = {231-247},
publisher = {Société mathématique de France},
title = {On the finiteness of Pythagoras numbers of real meromorphic functions},
url = {http://eudml.org/doc/272513},
volume = {138},
year = {2010},
}

TY - JOUR
AU - Acquistapace, Francesca
AU - Broglia, Fabrizio
AU - Fernando, José F.
AU - Ruiz, Jesús M.
TI - On the finiteness of Pythagoras numbers of real meromorphic functions
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 2
SP - 231
EP - 247
AB - We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real meromorphic power series. This measures the difficulty of the 17th Hilbert problem in the analytic case.
LA - eng
KW - 17th Hilbert problem; pythagoras number; sum of squares; bad set; germs at closed sets
UR - http://eudml.org/doc/272513
ER -

References

top
  1. [1] F. Acquistapace, F. Broglia, J. F. Fernando & J. M. Ruiz – « On the Pythagoras numbers of real analytic surfaces », Ann. Sci. École Norm. Sup.38 (2005), p. 751–772. Zbl1087.32003MR2195258
  2. [2] C. Andradas, L. Bröcker & J. M. Ruiz – Constructible sets in real geometry, Ergebn. Math. Grenzg., vol. 33, Springer, 1996. Zbl0873.14044MR1393194
  3. [3] C. Andradas, A. Díaz-Cano & J. M. Ruiz – « The Artin-Lang property for normal real analytic surfaces », J. reine angew. Math. 556 (2003), p. 99–111. Zbl1037.32010MR1971140
  4. [4] J. Bochnak, W. Kucharz & M. Shiota – « On equivalence of ideals of real global analytic functions and the 17th Hilbert problem », Invent. Math.63 (1981), p. 403–421. Zbl0467.32003MR620677
  5. [5] H. Cartan – « Variétés analytiques réelles et variétés analytiques complexes », Bull. Soc. Math. France85 (1957), p. 77–99. Zbl0083.30502MR94830
  6. [6] M. D. Choi, Z. D. Dai, T. Y. Lam & B. Reznick – « The Pythagoras number of some affine algebras and local algebras », J. reine angew. Math. 336 (1982), p. 45–82. Zbl0499.12018MR671321
  7. [7] C. N. Delzell – « A constructible continuous solution to Hilbert’s 17th problem, and other results in real Algebraic Geometry », Thèse, Stanford, 1980. MR2630715
  8. [8] J. F. Fernando – « On Hilbert’s 17th problem for global analytic functions in dimension 3 », Comment. Math. Helv.83 (2008), p. 67–100. Zbl1174.32007MR2365409
  9. [9] —, « On the positive extension property and Hilbert’s 17th problem for real analytic sets », J. reine angew. Math. 618 (2008), p. 1–49. Zbl1151.14040MR2404745
  10. [10] R. C. Gunning & H. Rossi – Analytic functions of several complex variables, Prentice-Hall Inc., 1965. Zbl0141.08601MR180696
  11. [11] P. Jaworski – « Extensions of orderings on fields of quotients of rings of real analytic functions », Math. Nachr.125 (1986), p. 329–339. Zbl0601.14018MR847371
  12. [12] —, « About estimates on number of squares necessary to represent a positive-definite analytic function », Arch. Math. (Basel) 58 (1992), p. 276–279. Zbl0723.14043MR1148203
  13. [13] J.-J. Risler – « Le théorème des zéros en géométries algébrique et analytique réelles », Bull. Soc. Math. France104 (1976), p. 113–127. Zbl0328.14001MR417167
  14. [14] J. M. Ruiz – « On Hilbert’s 17th problem and real Nullstellensatz for global analytic functions », Math. Z.190 (1985), p. 447–454. Zbl0579.14021MR806902
  15. [15] H. H. Schaefer – Topological vector spaces, Graduate Texts in Math., vol. 3, Springer, 1971. Zbl0217.16002MR342978

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.