Journal of Algebraic Combinatorics

, Volume 16, Issue 3, pp 231–237 | Cite as

Polynomials with All Zeros Real and in a Prescribed Interval

  • Jean B. Lasserre


We provide a characterization of the real-valued univariate polynomials that have only real zeros, all in a prescribed interval [a,b]. The conditions are stated in terms of positive semidefiniteness of related Hankel matrices.

algebraic combinatorics real algebraic geometry the \(\mathbb{K}\)-moment problem 


  1. 1.
    M. Aissen, I.J. Schoenberg, and A. Whitney, “On generating functions of totally positive sequences, I,” J. Analyse Math. 2 (1952), 93–103.Google Scholar
  2. 2.
    D.A. Beck, J.B. Remmel, and T. Whitehead, “The combinatorics of the transition matrices between the bases of the symmetric functions and the Bn analogues,” Discr. Math. 153 (1996), 3–27.Google Scholar
  3. 3.
    R.E. Curto and L. Fialkow, “The truncated complex K-moment problem,” Trans. Amer. Math. Soc. 352 (2000), 2825–2855.Google Scholar
  4. 4.
    F.R. Gantmacher, The Theory of Matrices: Vol II, Chelsea, New York, 1959.Google Scholar
  5. 5.
    J.B. Lasserre, “Polynomials with all zeros real and in a prescribed interval,” Technical Report, LAAS-CNRS, Toulouse, France, April 2001.Google Scholar
  6. 6.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1995.Google Scholar
  7. 7.
    D.D. Šiljak, Nonlinear Systems: The Parameter Analysis and Design, Wiley, New York, 1969.Google Scholar
  8. 8.
    D.D. Šiljak and M.D. Šiljak, “Nonnegativity of uncertain polynomials,” Mathematical Problems in Engineering 4 (1998), 135–163.Google Scholar
  9. 9.
    R.P. Stanley, “Positivity problems and conjectures in algebraic combinatorics,” in Mathematics: Frontiers and Perspectives, V. Arnold, M. Atiyah, P. Lax, and B. Mazur (Eds.), American Mathematical Society, Providence, RI, 2000, pp. 295–319.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Jean B. Lasserre
    • 1
  1. 1.LAAS-CNRSToulouse Cédex 4France

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