Polynomials with All Zeros Real and in a Prescribed Interval
Jean B. Lasserre
LAAS-CNRS 7 Avenue du Colonel Roche 31077 Toulouse Cédex 4 France
DOI: 10.1023/A:1021848304877
Abstract
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.
Pages: 231–237
Keywords: algebraic combinatorics; real algebraic geometry; the $\mathbb K$ \mathbb{k} -moment problem
Full Text: PDF
References
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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.
3. R.E. Curto and L. Fialkow, “The truncated complex K-moment problem,” Trans. Amer. Math. Soc. 352 (2000), 2825-2855.
4. F.R. Gantmacher, The Theory of Matrices: Vol II, Chelsea, New York, 1959.
5. J.B. Lasserre, “Polynomials with all zeros real and in a prescribed interval,” Technical Report, LAAS-CNRS, Toulouse, France, April 2001.
6. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1995.
7. D.D. \?Siljak, Nonlinear Systems : The Parameter Analysis and Design, Wiley, New York, 1969.
8. D.D. \?Siljak and M.D. \?Siljak, “Nonnegativity of uncertain polynomials,” Mathematical Problems in Engineering 4 (1998), 135-163.
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.