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Positive polynomials on nondegenerate basic semi-algebraic sets

  • Huy-Vui Ha and Toan Minh Ho EMAIL logo
Published/Copyright: October 13, 2016
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Advances in Geometry
From the journal Advances in Geometry Volume 16 Issue 4

Abstract

A concept of nondegenerate basic closed semi-algebraic sets in ℝn is introduced. These are unbounded closed semi-algebraic sets for which we obtain some representations of polynomials with positive infima (the polynomials are further assumed to be bounded if n>2) and solutions of the moment problem. The key to obtain these results is an explicit description of the algebra of bounded polynomials on a nondegenerate basic semi-algebraic set via the combinatorial information of the Newton polyhedron corresponding to the generators of the semi-algebraic set.

Keywords: Sum of squares; Positivstellensatz; Nichtnegativstellensatz; moment problem
MSC 2010: Primary 14P99; 44A60; Secondary 12E05; 12D15

Communicated by: C. Scheiderer

acknowledgement

We would like to thank the referee for the useful comments. This paper has been written while the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam. The authors would also like to thank the members in our group for their comments.

Funding

Research partially supported by NAFOSTED, Vietnam, Grant No. 101.04.2014.40, 101.04-2014.23.

References

[1] C. Berg, P. H. Maserick, Polynomially positive definite sequences. Math. Ann. 259 (1982), 487–495. MR660043 Zbl 0486.4400410.1007/BF01466054Search in Google Scholar

[2] J. Cimpric, S. Kuhlmann, C. Scheiderer, Sums of squares and moment problems in equivariant situations. Trans. Amer. Math. Soc. 361 (2009), 735–765. MR2452823 Zbl 1170.1404110.1090/S0002-9947-08-04588-1Search in Google Scholar

[3] S. Gindikin, L. R. Volevich, The method of Newton’s polyhedron in the theory of partial differential equations, volume 86 of Mathematics and its Applications (Soviet Series). Kluwer 1992. MR1256484 Zbl 0779.3500110.1007/978-94-011-1802-6Search in Google Scholar

[4] E. K. Haviland, On the Momentum Problem for Distribution Functions in More Than One Dimension. II. Amer. J. Math. 58 (1936), 164–168. MR1507139 Zbl 0015.10901JFM 62.0483.0110.2307/2371063Search in Google Scholar

[5] K. Kurdyka, M. Michalska, S. Spodzieja, Bifurcation values and stability of algebras of bounded polynomials. Adv. Geom. 14 (2014), 631–646. MR3276126 Zbl 1306.1402810.1515/advgeom-2014-0006Search in Google Scholar

[6] M. Marshall, Positive polynomials and sums of squares, volume 146 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2008. MR2383959 Zbl 1169.1300110.1090/surv/146Search in Google Scholar

[7] M. Marshall, Polynomials non-negative on a strip. Proc. Amer. Math. Soc. 138 (2010), 1559–1567. MR2587439 Zbl 1189.1406510.1090/S0002-9939-09-10016-3Search in Google Scholar

[8] M. Michalska, Algebras of bounded polynomials on unbounded semialgebraic sets. PhD thesis, Grenoble and Lodz 2011.Search in Google Scholar

[9] M. Michalska, Curves testing boundedness of polynomials on subsets of the real plane. J. Symbolic Comput. 56 (2013), 107–124. MR3061711 Zbl 1304.1407210.1016/j.jsc.2013.04.001Search in Google Scholar

[10] A. Nemethi, A. Zaharia, Milnor fibration at infinity. Indag. Math. (N.S.)3 (1992), 323–335. MR118 6741 Zbl 0806.5702110.1016/0019-3577(92)90039-NSearch in Google Scholar

[11] D. Plaumann, Sums of squares on reducible real curves. Math. Z. 265 (2010), 777–797. MR2652535 Zbl 1205.1407410.1007/s00209-009-0541-8Search in Google Scholar

[12] V. Powers, Positive polynomials and the moment problem for cylinders with compact cross-section. J. Pure Appl. Algebra188 (2004), 217–226. MR2030815 Zbl 1035.1402210.1016/j.jpaa.2003.10.009Search in Google Scholar

[13] V. Powers, C. Scheiderer, The moment problem for non-compact semialgebraic sets. Adv. Geom. 1 (2001), 71–88. MR1823953 Zbl 0984.4401210.1515/advg.2001.005Search in Google Scholar

[14] M. Putinar, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), 969–984. MR1254128 Zbl 0796.1200210.1512/iumj.1993.42.42045Search in Google Scholar

[15] C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245 (2003), 725–760. MR2020709 Zbl 1056.1407810.1007/s00209-003-0568-1Search in Google Scholar

[16] C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Math. 119 (2006), 395–410. MR2223624 Zbl 1120.1404710.1007/s00229-006-0630-5Search in Google Scholar

[17] C. Scheiderer, Positivity and sums of squares: a guide to recent results. In: Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math.Appl., 271–324, Springer 2009. MR2500469 Zbl 1156.1432810.1007/978-0-387-09686-5_8Search in Google Scholar

[18] K. Schmiidgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), 203–206. MR1092173 Zbl 0744.4400810.1007/978-3-319-64546-9_12Search in Google Scholar

[19] K. Schmiidgen, On the moment problem of closed semi-algebraic sets. J. Reine Angew. Math. 558 (2003), 225–234. MR1979186 Zbl 1047.4701210.1007/978-3-319-64546-9_13Search in Google Scholar

[20] M. Schweighofer, Global optimization of polynomials using gradient tentacles and sums of squares. SIAMJ. Optim. 17 (2006), 920–942. MR2257216 Zbl 1118.1302610.1137/050647098Search in Google Scholar

Received: 2012-11-2
Revised: 2015-8-19
Revised: 2016-2-23
Published Online: 2016-10-13
Published in Print: 2016-10-1

© 2016 by Walter de Gruyter Berlin/Boston

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https://doi.org/10.1515/advgeom-2016-0017
Keywords for this article
Sum of squares; Positivstellensatz; Nichtnegativstellensatz; moment problem

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  1. Research Article
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  3. Research Article
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