Sums of squares of regular functions on real algebraic varieties
HTML articles powered by AMS MathViewer
- by Claus Scheiderer
- Trans. Amer. Math. Soc. 352 (2000), 1039-1069
- DOI: https://doi.org/10.1090/S0002-9947-99-02522-2
- Published electronically: September 8, 1999
- PDF | Request permission
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 $1$.References
- Emil Artin, The collected papers of Emil Artin, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Edited by Serge Lang and John T. Tate. MR 0176888
- Ricardo Baeza, Quadratic forms over semilocal rings, Lecture Notes in Mathematics, Vol. 655, Springer-Verlag, Berlin-New York, 1978. MR 0491773
- 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
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- 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
- 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 Papers in Pure and Appl. Math., No. 46, Queen’s Univ., Kingston, Ont., 1977, pp. 385–405. MR 0498375
- 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, DOI 10.1016/0021-8693(80)90248-3
- 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, DOI 10.1016/0040-9383(95)00015-1
- 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
- Ch. N. Delzell: A constructive, continuous solution to Hilbert’s $17^\textrm {th}$ 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).
- 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.
- D. Gondard: Le 17ème problème de Hilbert et ses développements récents. Sém. Structures Algébriques Ordonnées, Univ. Paris VII, Vol. II, 21-49 (1990).
- Danielle Gondard and Paulo Ribenboim, Fonctions définies positives sur les variétés réelles, Bull. Sci. Math. (2) 98 (1974), no. 1, 39–47. MR 432614
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- 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.
- D. Hilbert: Über ternäre definite Formen. Acta math. 17, 169-197 (1893). See: Ges. Abh., Bd. II, Springer, Berlin, 1933, pp. 345-366.
- 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.
- 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.
- 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, DOI 10.1007/978-3-322-85033-1
- Hartmut Lindel, Projektive Moduln über Polynomringen $A[T_{1},\cdots ,T_{m}]$ mit einem regulären Grundring $A$, Manuscripta Math. 23 (1977/78), no. 2, 143–154 (German, with English summary). MR 472912, DOI 10.1007/BF01180569
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- 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, DOI 10.1007/978-1-4613-8655-1
- 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.
- 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
- 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, DOI 10.1017/CBO9780511526077
- Victoria Powers, Hilbert’s 17th problem and the champagne problem, Amer. Math. Monthly 103 (1996), no. 10, 879–887. MR 1427118, DOI 10.2307/2974612
- 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.
- Claus Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics, vol. 1588, Springer-Verlag, Berlin, 1994. MR 1321819, DOI 10.1007/BFb0074269
- Keying Guan and Shaofei Zhang, Structure of solvable subgroup of $\textrm {SL}(2,\mathbf C)$ and integrability of Fuchsian equations on torus $T^{2}$, Sci. China Ser. A 39 (1996), no. 5, 501–508. MR 1409807
- Konrad Schmüdgen, The $K$-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), no. 2, 203–206. MR 1092173, DOI 10.1007/BF01446568
- Gilbert Stengle, Integral solution of Hilbert’s seventeenth problem, Math. Ann. 246 (1979/80), no. 1, 33–39. MR 554130, DOI 10.1007/BF01352024
- E. Witt: Zerlegung reeller algebraischer Funktionen in Quadrate. Schiefkörper über reellem Funktionenkörper. J. reine angew. Math. 171, 4-11 (1934).
- Th. Wörmann: Positive polynomials on compact sets. To appear Manuscr. math.
Bibliographic Information
- Claus Scheiderer
- Affiliation: Fachbereich Mathematik, Universität Duisburg, 47048 Duisburg, Germany
- MR Author ID: 212893
- Email: claus.@math.uni-duisburg.de
- Received by editor(s): October 5, 1997
- Published electronically: September 8, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 1039-1069
- MSC (1991): Primary 14P99; Secondary 11E25, 12D15, 13H05, 14G30, 14H99, 14J99
- DOI: https://doi.org/10.1090/S0002-9947-99-02522-2
- MathSciNet review: 1675230
Dedicated: Dedicated to Manfred Knebusch on the occasion of his 60th birthday