Skip to main content
SpringerLink
Log in

Positivity and Sums of Squares: A Guide to Recent Results

  • Chapter
  • First Online:
Emerging Applications of Algebraic Geometry

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 149))

Abstract

This paper gives a survey, with detailed references to the literature, on recent developments in real algebra and geometry concerning the polarity between positivity and sums of squares. After a review of foundational material, the topics covered are Positiv- and Nichtnegativstellensatze, local rings, Pythagoras numbers, and applications to moment problems.

AMS(MOS) 2000 Subject Classifications. 14P05, l1E25, 44A60, 14P10.

On the occasion of the Algebraic Geometry tutorial, I spent a few inspiring days at the IMA at Minneapolis in April 2007. I would like to thank the institute for the kind invitation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. ART1N, Uber die Zerlegung definiter Funktioneii in Quadrate. Abh. Math. Sem. Univ. Hamburg 5:100-115 (1927).

    Google Scholar 

  2. E. BECKER AND V. POWE!U?, Sums of powers in rings and the real holomorphy ring. J. reine angew. Math. 480:71-103 (1996).

    Google Scholar 

  3. E. BECKER AND N. SCHWARTZ, Zum Darstellungssatz von Kadison-Dubois. Arch. Math. 40:421-428 (1983).

    Google Scholar 

  4. CH. BERG, J.P.R. CHRlSTENSEN, AND P. RESSEL, Positive definite functions on abelian semigroups. Math. Ann. 223:253-272 (1976).

    Google Scholar 

  5. R. BERR AND TH. WORMANN, Positive polynomials on compact sets. Manuscr. math. 104:135-143 (2001).

    Google Scholar 

  6. R. BERR AND TH. WORMANN, Positive polynomials and tame preorders. Math. Z. 236:813-840 (2001).

    Google Scholar 

  7. G. BLEKHERMAN, There are significantly more nonnegative polynomials than sums of squares. Israel J. Math. 153:355~380 (2006).

    Google Scholar 

  8. J. BOCHNAK, M. COSTE,. AND M.-F. Roy, Real Algebraic Geometry. Erg. Math. Grenzgeb. 36(3), Springer, Berlin, 1998.

    Google Scholar 

  9. L. BROCKER, Zur Theorie der quadratischen Formeti tiber formal reellen K orpern, Math. Ann. 210;233-256 (1974).

    Google Scholar 

  10. A. CAMP1LLO AND J. M. RUIZ, Some remarks on pythagorean real curve germs. J. Algebra 128:271-275 (1990).

    Google Scholar 

  11. M.D. CH01, Z.D. DA1, T.Y. LAM, AND B. REZNICK, The Pythagoras numberof some affine algebras and local algebras. J. reine angew. Math. 336: 45-82 (1982).

    Google Scholar 

  12. M.D. CHOI AND T.Y. LAM, An old question of Hilbert. In: Conf. Quadratic Forms (Kingston, Ont., 1976), edt G. Orzech, Queen's Papers Pure Appl. Math. 46:385-405 (1977).

    Google Scholar 

  13. J. CIMPRIC, S. KUHLMANN, AND C. SCHEIDERER, Sums of squares and moment problems in equivariant situations. Trans. Am. Math. Scc., 2007 (to appear).

    Google Scholar 

  14. CH.N. DELZELL, A constructive, continuous solution to Hilbert's 17t h problem, and other results in semi-algebraic geometry. Doctoral Dissertation, Stanford University, 1980.

    Google Scholar 

  15. J.F. FERNANDO, Positive semidefinite germs in real analytic surfaces. Math. Ann. 322:49--67 (2002).

    Google Scholar 

  16. J. F. FERNANDO, On the Pythagoras numbers of real analytic rings. J. Algebra. 243:321-338 (2001).

    Google Scholar 

  17. J. F. FERNANDO, Analytic surface germs with minimal Pythagoras number. Math. Z. 244;725-752 (2003); Erratum, ibid., 250:967-969 (2005).

    Google Scholar 

  18. J. F. FERNANDO AND J. M. RUIZ, Positive semidefinite germs on the cone. Pacific J. Math. 205:109-118 (2002).

    Google Scholar 

  19. J.F. FERNANDO, J.M. RU1Z, AND C. SCHEIDERER, Sums of squares in real rings. Trans. Am. Math. Soc. 356:2663-2684 (2003).

    Google Scholar 

  20. J.F. FERNANDO, J.M. RUIZ, AND C. SCHE1DERER, Sums of squares of linear forms. Math. Res. Lett. 13:947-956 (2006).

    Google Scholar 

  21. D. GR1MM, T. NETZER, AND M. SCHWEIGHOFER, A note on the representation of positive polynomials with structured sparsity. Preprint 2006.

    Google Scholar 

  22. D. HANDELMAN, Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132:35-62 (1988).

    Google Scholar 

  23. D. HANDELMAN, Polynomials with a positive power. In: Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math. 135, Am. Math. Soc., Providence, RI, 1992, pp. 229-230.

    Google Scholar 

  24. J.W. HELTON, S. MCCULLOUGH, M. PUTINAR, AND V. VINN1KOV, Convex matrix inequalities versus linear matrix inequalities. Preprint 2007.

    Google Scholar 

  25. J. W. HELTON AND M. PUTINAR, Positive polynomials in scalar and matrix variables, the spectral theorem and optimization. Preprint 2006.

    Google Scholar 

  26. D. HILBERT, tiber die Darstellung definiter Formen als Summe von Formetiquadraten. Math. Ann. 32:342-350 (1888).

    Google Scholar 

  27. D. HILBERT, tiber terruire definite Formen. Acta math. 17:169-197 (1893).

    Google Scholar 

  28. TH. JACOBI, A representation theorem for certain partially ordered commutative rings. Math. Z. 237:259-273 (2001).

    Google Scholar 

  29. TH. JACOBI, Uber die Darstellung positives: Polynome auf semi-algebraischen Kompakta. Doctoral Dissertation, Universitat Konstanz, 1999.

    Google Scholar 

  30. TH. JACOBI AND A. PRESTEL, Distinguished representations of strictly positive polynomials. J. reine angew. Math. 532:223-235 (2001).

    Google Scholar 

  31. M. KNEBUSCH AND C. SCHEIDERER, Einfiihrung in die reelle Algebra. Vieweg, Wiesbaden, 1989.

    Google Scholar 

  32. J. KOLLAR, Sharp effective Nullstellensatz. J. Am. Math. Soc. 1:963-975 (1988).

    Google Scholar 

  33. J. L. KRIVINE, Anneaux preordonnes. J. Analyse Math. 12:307-326 (1964).

    Google Scholar 

  34. S. KUHLMANN AND M. MARSHALL, Positivity, sums of squares and the multidimensional moment problem. Trans. Am. Math. Soc. 354:4285-4301 (2002).

    Google Scholar 

  35. S. KUHLMANN, M. MARSHALL, AND N. SCHWARTZ, Positivity, sums of squares and the multi-dimensional moment problem II. Adv. Geom. 5:583--606 (2005).

    Google Scholar 

  36. S. KUHLMANN AND M. PUTINAR, Positive polynomials on projective limits of real algebraic varieties. Preprint, IMA Preprint Series 2162, 2007.

    Google Scholar 

  37. J.B. LASSERRE, A sum of squares approximation of nonnegative polynomials. SIAM J. Optim. 16:751-765 (2006).

    Google Scholar 

  38. J.B. LASSERRE, Sum of squares approximation of polynomials, nonnegative on a real algebraic set. SIAM J. Optim. 16:610--628 (2006).

    Google Scholar 

  39. J. B. LASSERRE, Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17:822-843 (2006).

    Google Scholar 

  40. J.B. LASSERRE AND T. NETZER, SOS approximations of nonnegative polynomials via simple high degree perturbations. Math. Z. 256:99-112 (2007).

    Google Scholar 

  41. M. LAURENT, Sums of squares, moment matrices and optimization over polynomials. This volume.

    Google Scholar 

  42. J.A. DE LOERA AND F. SANTOS, An eff~ctive version of P6lya's theorem on positive definite forms. J. Pure Applied Algebra 108:231-240 (1996); Erratum, ibid. 155:309~310 (2001).

    Google Scholar 

  43. L. MARE, Level and Pythagoras number of some geometric rings. Math. Z. 204:615-629 (1990); Erratum, ibid. 209:481-483 (1992).

    Google Scholar 

  44. M. MARSHALL, A real holomorphy ring without the Schmiidgen property. Canad, Math. Bull. 42:354-358 (1999).

    Google Scholar 

  45. M. MARSHALL, Positive Polynomials and Sums of Squares. Istituti Editoriali e Poligrafici Internazion ali, Pisa, 2000.

    Google Scholar 

  46. M. MARSHALL, Extending the archimedeari Positivstellensatz to the noncompact case. Canad. Math. Bull. 44:223-230 (2001).

    Google Scholar 

  47. M. MARSHALL, A general representation theorem for partially ordered commutative rings. Math. Z. 242:217-225 (2002).

    Google Scholar 

  48. M. MARSHALL, Representation of non-negative polynomials with finitely many zeros. Ann. Fac. Sci. Toulouse (6) 15:599--609 (2006).

    Google Scholar 

  49. J. -PH. MONNIER, Anneaux d'holomorphie ei Positivstellensatz archimedien, Manuscr, Math. 97:269-302 (1998).

    Google Scholar 

  50. T. NETZER, An elementary proof of Schmiidgen's theorem on the moment problem of closed semi-algebraic sets. Preprint, 2006.

    Google Scholar 

  51. J. ORTEGA, On the Pythagoras number of a real irreducible algebroid curve. Math. Ann. 289:111-123 (1991).

    Google Scholar 

  52. A. PFISTER, Quadratic Forms with Applications to Algebraic Geometry and Topology. London Math. Soc. Lect. Notes 217, Cambridge, 1995.

    Google Scholar 

  53. S. PINGEL, Der reelle Holomorphiering von Algebren. Doctoral Dissertation, Fernuniversitat Hagen, 1998.

    Google Scholar 

  54. D. PLAUMANN, Bounded polynomials, sums of squares and the moment problem. Doctoral dissertation, Univ. Konstanz, 2008.

    Google Scholar 

  55. V. POWERS AND B. REZNICK, Polynomials that are positive on an interval. Trans. Am. Math. Soc. 352:4677-4592 (2000).

    Google Scholar 

  56. V. POWERS AND B. REZNICK, A new bound for P6lya's theorem with applications to polynomials positive on polyhedra. J. Pure Applied Algebra 164;221-229 (2001).

    Google Scholar 

  57. V. POWERS AND C. SCHEIDERER, The moment problem for non-compact semialgebraic sets. Adv. Geom. 1:71-88 (2001).

    Google Scholar 

  58. A. PRESTEL, Lectures on Formally Real Fields. Lect. Notes Math. 1093, Springer, Berlin, 1984.

    Google Scholar 

  59. A. PRESTEL AND CH.N. DELZELL, Positive Polynomials. Monographs in Mathematics, Springer, Berlin, 2001.

    Google Scholar 

  60. M. PUTINAR, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42:969~984 (1993).

    Google Scholar 

  61. M. PUT1NAR AND C. SCHEIDERER, Multivariate moment problems.' Geometry and indeterminateness. Ann. Sc. Norm. Pisa (5) 5:137-157 (2006).

    Google Scholar 

  62. M. PUT1NAR AND F.-H. VASILESCU, Solving moment problems by dimensional extension. Ann. Math. 149:1097-1107 (1999).

    Google Scholar 

  63. R. QUAREZ, Pythagoras numbers of real algebroid curves and Gram matrices. J. Algebra 238:139-158 (2001).

    Google Scholar 

  64. B. REZNICK, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220:75-97 (1995).

    Google Scholar 

  65. B. REZNICK, Some concrete aspects of Hilbert's 17th problem. Proe. RAGOS Conf., Contemp. Math. 253:251-272 (2000).

    Google Scholar 

  66. B. REZNICK, On the absence of uniform denominators in Hilbert's 17th problem. Proc. Am. Math. Soc. 133:2829-2834 (2005).

    Google Scholar 

  67. B. REZNICK, On Hilbert's construction of positive polynomials. Preprint, 2007.

    Google Scholar 

  68. J.M. RUIZ, Sums of two squares in analytic rings. Math. Z. 230:317-328 (1999).

    Google Scholar 

  69. W. SCHARLAU, Quadratic and Hermitian Forms. Grundl. Math. Wiss. 270, Springer, Berlin, 1985.

    Google Scholar 

  70. C. SCHEIDERER, Real algebra and its applications to geometry in the last ten years: Some major developments and results. In: Real algebraic geometry (Rennes 1991), M. Coste, L. Mahe, M.-F. Roy (eds.), Lect. Notes Math. 1524, Springer, Berlin, 1992, pp. 75-96.

    Google Scholar 

  71. C. SCHEIDERER, Sums of squares of regular functions on real algebraic varieties. Trans. Am. Math. Soc. 352:1039-1069 (1999).

    Google Scholar 

  72. C. SCHEIDERER, On sums of squares in local rings. J. reine angew, Math. 540:205-227 (2001).

    Google Scholar 

  73. C. SCHEIDERER, Sums of squares on real algebraic curves. Math. Z. 245:725760 (2003).

    Google Scholar 

  74. C. SCHEIDERER, Distinguished representations of non-negative polynomials. J. Algebra 289:558-573 (2005).

    Google Scholar 

  75. C. SCHEIDERER, Non-existence of degree bounds for weighted sums of squares representations. J. Complexity 21:823-844 (2005).

    Google Scholar 

  76. C. SCHEIDERER, Sums of squares on real algebraic surfaces. Manuscr. math. 119:395-410 (2006).

    Google Scholar 

  77. C. SCHEIDERER, Local study of preorderings (in preparation).

    Google Scholar 

  78. J. SCHMID, On the degree complexity of Hilbert's 17th problem and the real Nullstellensatz. Habilitationsschrift, Univsrsitat Dortmund, 1998.

    Google Scholar 

  79. K. SCHMUDGEN, The K -mometii problem [or compact semi-algebraic sets. Math. Ann. 289:203-206 (1991).

    Google Scholar 

  80. K. SCHMUDGEN, On the moment problem of closed semi-algebraic sets. J. reine angew. Math. 558;225-234 (2003).

    Google Scholar 

  81. K. SCHMUDGEN, Noncommutative real algebraic geometry - some basic concepts and first ideas. This volume.

    Google Scholar 

  82. M. SCHWEIGHOFER, An algorithmic approach to Schmiidgen's Positivstellensaiz, J. Pure Applied Algebra 166:307-319 (2002).

    Google Scholar 

  83. M. SCHWEIGHOFER, Iterated rings of bounded elements and generalizations of Schmudgen's Positivstellensatz. J. reine angew. Math. 554:19-45 (2003).

    Google Scholar 

  84. M. SCHWEIGHOFER, On the complexity of Schmiidgen's Positivstellensatz. J. Complexity 20:529-543 (2004).

    Google Scholar 

  85. M. SCHWEIGHOFER, Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets. Manuser. math. 117:407-428 (2005).

    Google Scholar 

  86. M. SCHWEIGHOFER, Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17:920-942 (2006).

    Google Scholar 

  87. G. STENGLE, A nullstellensatz and a positivstellensatz in semiolqebraic geometry. Math. Ann. 207:87-97 (1974).

    Google Scholar 

  88. G. STENGLE~ Complexity estimates for the Schmiidgen Positivstellensatz. J. Complexity 12:167-174 (1996).

    Google Scholar 

  89. B. STURMFELS, Solving Systems of Polynomial Equations. CBMS Reg. Conf. Ser. Math., Am. Math. Soc., Providence, RI, 2002.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Universität Konstanz, 78457, Konstanz, Germany

    Claus Scheiderer

Authors
  1. Claus Scheiderer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claus Scheiderer .

Editor information

Editors and Affiliations

  1. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

  2. Department of Mathematics Harvard University, Cambridge, 02138, U.S.A.

    Seth Sullivant

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Scheiderer, C. (2009). Positivity and Sums of Squares: A Guide to Recent Results. In: Putinar, M., Sullivant, S. (eds) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09686-5_8

Download citation

Publish with us

Policies and ethics

Search

Navigation