Positive polynomials on nondegenerate basic semi-algebraic sets

Huy-Vui Ha 1  and Toan Minh Ho 1
  • 1 Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Ha Noi, Viet Nam
Huy-Vui Ha and Toan Minh Ho

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.

  • [1]

    C. Berg, P. H. Maserick, Polynomially positive definite sequences. Math. Ann. 259 (1982), 487–495. MR660043 Zbl 0486.44004

    • Crossref
    • Export Citation
  • [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 1170.14041

  • [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 0779.35001

  • [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 62.0483.01

    • Crossref
    • Export Citation
  • [5]

    K. Kurdyka, M. Michalska, S. Spodzieja, Bifurcation values and stability of algebras of bounded polynomials. Adv. Geom. 14 (2014), 631–646. MR3276126 Zbl 1306.14028

  • [6]

    M. Marshall, Positive polynomials and sums of squares, volume 146 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2008. MR2383959 Zbl 1169.13001

  • [7]

    M. Marshall, Polynomials non-negative on a strip. Proc. Amer. Math. Soc. 138 (2010), 1559–1567. MR2587439 Zbl 1189.14065

  • [8]

    M. Michalska, Algebras of bounded polynomials on unbounded semialgebraic sets. PhD thesis, Grenoble and Lodz 2011.

  • [9]

    M. Michalska, Curves testing boundedness of polynomials on subsets of the real plane. J. Symbolic Comput. 56 (2013), 107–124. MR3061711 Zbl 1304.14072

    • Crossref
    • Export Citation
  • [10]

    A. Nemethi, A. Zaharia, Milnor fibration at infinity. Indag. Math. (N.S.)3 (1992), 323–335. MR118 6741 Zbl 0806.57021

    • Crossref
    • Export Citation
  • [11]

    D. Plaumann, Sums of squares on reducible real curves. Math. Z. 265 (2010), 777–797. MR2652535 Zbl 1205.14074

    • Crossref
    • Export Citation
  • [12]

    V. Powers, Positive polynomials and the moment problem for cylinders with compact cross-section. J. Pure Appl. Algebra188 (2004), 217–226. MR2030815 Zbl 1035.14022

    • Crossref
    • Export Citation
  • [13]

    V. Powers, C. Scheiderer, The moment problem for non-compact semialgebraic sets. Adv. Geom. 1 (2001), 71–88. MR1823953 Zbl 0984.44012

  • [14]

    M. Putinar, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), 969–984. MR1254128 Zbl 0796.12002

    • Crossref
    • Export Citation
  • [15]

    C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245 (2003), 725–760. MR2020709 Zbl 1056.14078

    • Crossref
    • Export Citation
  • [16]

    C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Math. 119 (2006), 395–410. MR2223624 Zbl 1120.14047

    • Crossref
    • Export Citation
  • [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 1156.14328

  • [18]

    K. Schmiidgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), 203–206. MR1092173 Zbl 0744.44008

    • Crossref
    • Export Citation
  • [19]

    K. Schmiidgen, On the moment problem of closed semi-algebraic sets. J. Reine Angew. Math. 558 (2003), 225–234. MR1979186 Zbl 1047.47012

  • [20]

    M. Schweighofer, Global optimization of polynomials using gradient tentacles and sums of squares. SIAMJ. Optim. 17 (2006), 920–942. MR2257216 Zbl 1118.13026

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Search