Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets

Manuscripta Mathematica

Volumn 117, Issue 4, 2005, Pages 407-428

  Schweighofer, Markus ^a  

^a UNIVERSITY OF KONSTANZ   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23944437404     PISSN: 00252611     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00229-005-0568-z     Document Type: Article

References (32)

1. [AT] de Angelis, V., Tuncel, S.: Handelman's theorem on polynomials with positive multiples. In: Marcus, Brian et al. (ed.) Codes, systems, and graphical models. IMA Vol. Math. Appl. 123, 439-445 (2001)
2. [BCR] Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 36, Springer, Berlin, 1998
3. [BLP] Bonsall, F., Lindenstrauss, J., Phelps, R.: Extreme positive operators on algebras of functions. Math. Scand. 18, 161-182 (1966)
4. [BW] Berr, R., Wörmann, T.: Positive polynomials and tame preorderings. [J] Math. Z. 236 (4), 813-840 (2001)
5. [H1] Handelman, D.: Positive polynomials and product type actions of compact groups. Mem. Am. Math. Soc. 320 (1985)
6. [H2] Handelman, D.: Deciding eventual positivity of polynomials. Ergodic Theory Dyn. Syst. 6, 57-79 (1986)
7. [H3] Handelman, D.: Positive polynomials, convex integral polytopes, and a random walk problem. Lecture Notes in Mathematics, Springer, Berlin, 1282, 1987
8. [H4] Handelman, D.: Representing polynomials by positive linear functions on compact convex polyhedra. Pac. J. Math. 132 (1), 35-62 (1988)
9. [H5] Handelman, D.: Polynomials with a positive power, Symbolic dynamics and its applications. Proc. AMS Conf. in honor of R. L. Adler, New Haven/CT (USA) 1991, Contemp. Math. 135, 229-230 (1992)
10. [Har] Harrison, D.: Finite and infinite primes for rings and fields. Mem. Am. Math. Soc. 68 (1966)
11. [Jac] Jacobi, T.: A representation theorem for certain partially ordered commutative rings. Math. Z. 237 (2), 259-273 (2001)
12. [Jam] Jameson, G.: Ordered linear spaces Lecture Notes in Mathematics 141, Springer, Berlin-Heidelberg-New York, 1970
13. [KMS] Kuhlmann, S., Marshall, M., Schwartz, N.: Positivity, sums of squares and the multi-dimensional moment problem II. submitted
14. [Köt] Köthe, G.: Topological vector spaces I. Springer, Berlin-Heidelberg-New York, 1969
15. [L1] Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 (3), 796-817 (2001)
16. [L2] Lasserre, J.: Polynomial programming: LP-relaxations also converge. SIAM J. Opt. 15, 383-393 (2005)
17. [Mar] Marshall, M.: Representation of non-negative polynomials having finitely many zeros, to appear in Annales de la Faculté des Sciences de Toulouse http:// math.usask.ca/~marshall/
18. [PD] Prestel, A., Delzell, C.: Positive polynomials. Springer Monographs in Mathematics, Springer, Berlin, 2001
19. [Pól] Pólya, G.: Über positive Darstellung von Polynomen. Vierteljahresschrift der Naturforschenden Gesellschaft in Zürich 73 (1928), 141-145,
20. reprinted in: Collected Papers, Volume 2, 309-313, Cambridge: MIT Press (1974)
21. [PR] Powers, V., Reznick, B.: A new bound for Pólya's theorem with applications to polynomials positive on polyhedra. J. Pure Appl. Algebra 164 (1-2), 221-229 (2001)
22. [Put] Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (3), 969-984 (1993)
23. [PV] Putinar, M., Vasilescu, F.-H.: Solving moment problems by dimensional extension. Ann. Math. (2) 149 (3), 1087-1107 (1999)
24. [Rez] Reznick, B.: Some concrete aspects of Hilbert's 17th problem. In: Delzell, Charles N. et al. (ed.) Real algebraic geometry and ordered structures. Contemp. Math. 253, 251-272 (2000)
25. [S1] Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245 (4), 725-760 (2003)
26. [S2] Scheiderer, C.: Distinguished representations of non-negative polynomials. preprint http://www.uni-duisburg.de/FB11/FGS/F1/claus. html#preprints
27. [S3] Scheiderer, C.: Sums of squares on real algebraic surfaces. preprint http:// www.uni-duisburg.de/FB11/FGS/F1/claus.html#preprints
28. [Sch] Schmüdgen, K.: The K -moment problem for compact semi-algebraic sets. Math. Ann. 289 (2), 203-206 (1991)
29. [Sw1] Schweighofer, M.: An algorithmic approach to Schmüdgen's Positivstellensatz. J. Pure Appl. Algebra 166 (3), 307-319 (2002)
30. [Sw2] Schweighofer, M.: Iterated rings of bounded elements and generalizations of Schmüdgen's Positivstellensatz. J. Reine Angew. Math. 554, 19-45 (2003)
31. [Sw3] Schweighofer, M.: On the complexity of Schmüdgen's Positivstellensatz. J. Complexity 20, 529-543 (2004)
32. [Sw4] Schweighofer, M.: Optimization of polynomials on compact semialgebraic sets, to appear in SIAM J. Opt., 15 (3), 805-825 (2005) http://www.mathe.uni-konstanz.de/homepages/schweigh/