Abstract
In a variety of systems theory problems ranging from tests from Lyapunov stability, existence of limit cycles in nonlinear systems, existence of an operating point for a nonlinear circuit, the output feedback stabilization problem, multidimensional filter stability tests (see Chap. 4), tests for multivariate positive realness in electrical network realizability theory, etc., it is required to test a specified polynomial in several real variables for global or nonglobal positivity (nonnegativity). The topics of this chapter, dealing with the question of existence of such tests, followed by their actual construction depend a lot on the results of elementary decision algebra. Section 2.2 is, therefore, devoted to a concise exposition of the theory underlying the procedure for deciding the solvability in a real-closed field of a finite system of polynomial equations and inequalities with rational coefficients. For the development of this topic as a part of mathematical logic, the reader is referred to [30]. Here the discussion of the foregoing topic is centered around the quantifier elimination algorithm of Tarski [30] using a notion introduced by Cohen [31], the method of Seidenberg using the theory of resultants [32], and a cylindrical algebraic decomposition scheme for quantifier elimination developed by Collins in 1975 [2]. This last technique was also independently pursued by Bose and Modarressi, who used the theory of resultants to develop an algorithm for testing a multivariate polynomial for global positivity as reported in 1976 [2].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
N. K. Bose, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, New York, 1982.
A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, CA., 1930.
P. J. Cohen, “Decision procedures for real and p-adic fields,” Comm. Pure Appl. Math, vol. 22, pp. 131–151, 1969.
A. Seidenberg, “A new decision method for elementary algebra,” Ann. Math., vol. 60, pp. 365–374, 1954.
N. Jacobson, Lectures in Abstract Algebra, vol. 3, Springer-Verlag, New York, 1964.
E. Engeler, Introduction to the Theory of Computation, Academic Press, New York, 1973.
J. M. Thomas, “Sturm’s theorem for multiple roots,” Nat. Math. Mag., vol. 15, pp. 391–394, 1941.
B. D. O. Anderson, N. K. Bose, and E. I. Jury, “Output feedback stabilization and related problems - solution via decision methods,” IEEE Trans. Auto. Control, vol. 20, pp. 53–66, Feb. 1975.
E. Zerz, Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences 256. Springer-Verlag London Limited, London, Great Britain, 2000.
P. A. Parrilo, “Semidefinite programming relaxations for semialgebraic problems,” Math. Prog. Ser. B, vol. 96, no. 2, pp. 293–320, 2003.
C. N. Delzell, A Constructive Continuous Solution to Hilbert’s 17th Problem and Other Results in Semi-Algebraic Geometry, Ph.D. thesis, Stanford University, Math. Dept., June 1980.
D. Henrion and A. Garulli (eds.), Positive Polynomials in Control, Springer-Verlag Berlin Heidelberg, 2005.
N. Z. Shor, “Class of global minimum bounds of polynomial functions,” Cybernetics, vol. 23, pp. 731–734, 1987.
Bose, N., and C. Li. “A quadratic form representation of polynomials of several variables and its applications.” IEEE Transactions on Automatic Control 13.4 (1968): 447–448.
Parrilo, P. A. (2000) Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization (Doctoral dissertation, California Institute of Technology).
P. Parrilo, “Exploiting algebraic structure in sum of squares programs,” in Positive polynomials in control, D. Henrion and A. Garulli, Eds., pp. 181–194. Springer, 2005.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Bose, N.K. (2017). Multivariate Polynomial Positivity (Nonnegativity) Tests. In: Applied Multidimensional Systems Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-46825-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-46825-9_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46824-2
Online ISBN: 978-3-319-46825-9
eBook Packages: EngineeringEngineering (R0)