Recent Advances on Determining the Number of Real Roots of Parametric Polynomials

An explicit criterion for the determination of the numbers and multiplicities of the real/imaginary roots for polynomials with symbolic coefficients is based on a Complete Discrimination System (CDS). A CDS is a set of explicit expressions in terms of the coefficients that are sufficient for determining the numbers and multiplicities of the real and imaginary roots. Basically, the problem is considered on a total real axis and a total complex plane. However, it is often required in both practice and theory to determine the number of real roots in some interval, especially (0,∞ ) or (?∞, 0). This article is mainly devoted to solving the case in an interval, but some global results are reviewed for understanding. It is shown, with examples, how useful the CDS can be in order to understand the behaviour of the roots of an univariate polynomial in terms of the coefficients.

[1]  W. Habicht.  Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens , 1948 .

[2]  Bruno Buchberger,et al.  Computer algebra symbolic and algebraic computation , 1982, SIGS.

[3]  R. Loos.  Generalized Polynomial Remainder Sequences , 1983 .

[4]  Scott F. Smith,et al.  Towards mechanical solution of the Kahan Ellipse Problem 1 , 1983, EUROCAL.

[5]  Maurice Mignotte,et al.  On Mechanical Quantifier Elimination for Elementary Algebra and Geometry , 1988, J. Symb. Comput..

[6]  Dennis S. Arnon,et al.  Geometric Reasoning with Logic and Algebra , 1988, Artif. Intell..

[7]  Daniel Lazard,et al.  Quantifier Elimination: Optimal Solution for Two Classical Examples , 1988, J. Symb. Comput..

[8]  Bruce W. Char,et al.  Maple V Library Reference Manual , 1992, Springer New York.

[9]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[10]  David W. Lewis,et al.  Matrix theory , 1991 .

[11]  Volker Weispfenning,et al.  Quantifier elimination for real algebra—the cubic case , 1994, ISSAC '94.

[12]  Lu Yang,et al.  A complete discrimination system for polynomials , 1996 .

[13]  F. Broglia.  Lectures in real geometry , 1996 .

[14]  Lu Yang,et al.  Explicit Criterion to Determine the Number of Positive Roots of a Polynomial 1) , 1997 .

[15]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[16]  L. González-Vega.  A Combinatorial Algorithm Solving Some Quantifier Elimination Problems , 1998 .

[17]  Marie-Françoise Roy,et al.  Sturm—Habicht Sequences, Determinants and Real Roots of Univariate Polynomials , 1998 .

[18]  S. Liang,et al.  A complete discrimination system for polynomials with complex coefficients and its automatic generation , 1999 .