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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Sums of squares of regular functions
on real algebraic varieties


Author: Claus Scheiderer
Journal: Trans. Amer. Math. Soc. 352 (2000), 1039-1069
MSC (1991): Primary 14P99; Secondary 11E25, 12D15, 13H05, 14G30, 14H99, 14J99
Published electronically: September 8, 1999
MathSciNet review: 1675230
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Abstract: Let $V$ be an affine algebraic variety over $\mathbb{R}$ (or any other real closed field $R$). We ask when it is true that every positive semidefinite (psd) polynomial function on $V$ is a sum of squares (sos). We show that for $\dim V\ge 3$ the answer is always negative if $V$ has a real point. Also, if $V$ is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if $V$ is a smooth surface with only real divisors at infinity. The ``compact'' case is harder. We completely settle the case of smooth curves of genus $\le 1$: If such a curve has a complex point at infinity, then every psd function is sos, provided the field $R$ is archimedean. If $R$ is not archimedean, there are counter-examples of genus $1$.


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  • 1. Emil Artin, The collected papers of Emil Artin, Edited by Serge Lang and John T. Tate, Addison–Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. MR 0176888
  • 2. Ricardo Baeza, Quadratic forms over semilocal rings, Lecture Notes in Mathematics, Vol. 655, Springer-Verlag, Berlin-New York, 1978. MR 0491773
  • 3. J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 12, Springer-Verlag, Berlin, 1987 (French). MR 949442
  • 4. Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
  • 5. 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, 10.1515/crll.1982.336.45
  • 6. Man Duen Choi and Tsit Yuen Lam, An old question of Hilbert, Conference on Quadratic Forms—1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976) Queen’s Univ., Kingston, Ont., 1977, pp. 385–405. Queen’s Papers in Pure and Appl. Math., No. 46. MR 0498375
  • 7. M. D. Choi, T. Y. Lam, B. Reznick, and A. Rosenberg, Sums of squares in some integral domains, J. Algebra 65 (1980), no. 1, 234–256. MR 578805, 10.1016/0021-8693(80)90248-3
  • 8. J.-L. Colliot-Thélène and C. Scheiderer, Zero-cycles and cohomology on real algebraic varieties, Topology 35 (1996), no. 2, 533–559 (English, with English and French summaries). MR 1380515, 10.1016/0040-9383(95)00015-1
  • 9. Michel Coste and Marie-Françoise Roy, La topologie du spectre réel, Ordered fields and real algebraic geometry (San Francisco, Calif., 1981), Contemp. Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 27–59 (French). MR 653174
  • 10. Ch. N. Delzell: A constructive, continuous solution to Hilbert's $17^{\rm th}$ problem, and other results in semi-algebraic geometry. Ph. D. thesis, Stanford University, June 1980. Cf. also ``Bad points for positive semidefinite polynomials: preliminary report'', Abstracts of papers presented to the AMS 18, # 926-12-174 (1997).
  • 11. Ch. N. Delzell: Kreisel's unwinding of Artin's proof. In: Kreiseliana: About and Around Georg Kreisel, P. Odifreddi (ed.), A. K. Peters, Wellesley, MA, 1996, pp.113-246. CMP 97:08
  • 12. D. Gondard: Le 17ème problème de Hilbert et ses développements récents. Sém. Structures Algébriques Ordonnées, Univ. Paris VII, Vol. II, 21-49 (1990).
  • 13. Danielle Gondard and Paulo Ribenboim, Fonctions définies positives sur les variétés réelles, Bull. Sci. Math. (2) 98 (1974), no. 1, 39–47. MR 0432614
  • 14. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • 15. D. Hilbert: Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32, 342-350 (1888). See: Ges. Abh., Bd. II, Springer, Berlin, 1933, pp.154-161.
  • 16. D. Hilbert: Über ternäre definite Formen. Acta math. 17, 169-197 (1893). See: Ges. Abh., Bd. II, Springer, Berlin, 1933, pp. 345-366.
  • 17. D. Hilbert: Mathematische Probleme. Arch. Math. Phys. (3) 1, 44-63 and 213-237 (1901). See: Ges. Abh., Bd. III, Springer, Berlin, 1933, pp. 290-329.
  • 18. D. Hilbert: Hermann Minkowski. Gedächtnisrede, 1. Mai 1909. Math. Ann. 68, 445-471 (1910). See: Ges. Abh., Bd. III, Springer, Berlin, 1933, pp. 339-364.
  • 19. Manfred Knebusch and Claus Scheiderer, Einführung in die reelle Algebra, Vieweg Studium: Aufbaukurs Mathematik [Vieweg Studies: Mathematics Course], vol. 63, Friedr. Vieweg & Sohn, Braunschweig, 1989 (German). MR 1029278
  • 20. Hartmut Lindel, Projektive Moduln über Polynomringen 𝐴[𝑇₁,\cdots,𝑇_{𝑚}] mit einem regulären Grundring 𝐴, Manuscripta Math. 23 (1977/78), no. 2, 143–154 (German, with English summary). MR 0472912
  • 21. Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • 22. Gary Cornell and Joseph H. Silverman (eds.), Arithmetic geometry, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR 861969
  • 23. H. Minkowski: Untersuchungen über quadratische Formen. Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enthält. Inauguraldissertation, Königsberg 1885; see Ges. Abh., Bd. I, Teubner, Leipzig, 1911, pp. 157-202.
  • 24. T. S. Motzkin, The arithmetic-geometric inequality, Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, New York, 1967, pp. 205–224. MR 0223521
  • 25. Albrecht Pfister, Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Note Series, vol. 217, Cambridge University Press, Cambridge, 1995. MR 1366652
  • 26. Victoria Powers, Hilbert’s 17th problem and the champagne problem, Amer. Math. Monthly 103 (1996), no. 10, 879–887. MR 1427118, 10.2307/2974612
  • 27. B. Reznick: Some concrete aspects of Hilbert's 17th problem. Preprint, see Sém. Structures Algébriques Ordonnées, Univ. Paris VII, 1996. Revised version to appear in Proc. RAGOS, Contemp. Math.
  • 28. Claus Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics, vol. 1588, Springer-Verlag, Berlin, 1994. MR 1321819
  • 29. Keying Guan and Shaofei Zhang, Structure of solvable subgroup of 𝑆𝐿(2,𝐂) and integrability of Fuchsian equations on torus 𝐓², Sci. China Ser. A 39 (1996), no. 5, 501–508. MR 1409807
  • 30. Konrad Schmüdgen, The 𝐾-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), no. 2, 203–206. MR 1092173, 10.1007/BF01446568
  • 31. Gilbert Stengle, Integral solution of Hilbert’s seventeenth problem, Math. Ann. 246 (1979/80), no. 1, 33–39. MR 554130, 10.1007/BF01352024
  • 32. E. Witt: Zerlegung reeller algebraischer Funktionen in Quadrate. Schiefkörper über reellem Funktionenkörper. J. reine angew. Math. 171, 4-11 (1934).
  • 33. Th. Wörmann: Positive polynomials on compact sets. To appear Manuscr. math.

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Additional Information

Claus Scheiderer
Affiliation: Fachbereich Mathematik, Universität Duisburg, 47048 Duisburg, Germany
Email: claus.@math.uni-duisburg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02522-2
Keywords: Sums of squares, positive semidefinite functions, preorders, real algebraic curves, Jacobians, real algebraic surfaces, real spectrum
Received by editor(s): October 5, 1997
Published electronically: September 8, 1999
Dedicated: Dedicated to Manfred Knebusch on the occasion of his 60th birthday
Article copyright: © Copyright 1999 American Mathematical Society