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**A recursive algorithm for constructing generalized Sturm sequence.**
*(English)*
Zbl 0984.12008

As is known, a generalized Sturm sequence (GSS), as well as the classical Sturm sequence, plays a very important role in real roots classification of algebraic equations, but it is very difficult to compute it with symbolic coefficients.

L. González-Vega, T. Reico, H. Lombardi and M.-F. Roy [Caviness (ed.) et al., Quantifier elimination and cylindrical algebraic decomposition, 300-316 (1998; Zbl 0900.12002)] introduced the Sturm-Habicht sequence which generalized the work of W. Habicht [Comment. Math. Helv. 21, 99-116 (1948; Zbl 0029.24402)]. Although the above sequence is easier to compute compared with the conventional approach, such a sequence is not the Sturm sequence in the usual sense because there might be zero-polynomials in this sequence.

In this paper, the authors use a recursive method to construct GSS for two parametric polynomials via the subresultant polynomial chain of the two polynomials avoiding complicated division operations. They establish a connection between GSS and the subresultant polynomial chain of two polynomials and an algorithm that computes the leading coefficients and degrees of the GSS for the infinite interval case. The algorithm has been implemented in Maple and is efficient for constructing the leading terms of the GSS for polynomials with symbolic coefficients.

L. González-Vega, T. Reico, H. Lombardi and M.-F. Roy [Caviness (ed.) et al., Quantifier elimination and cylindrical algebraic decomposition, 300-316 (1998; Zbl 0900.12002)] introduced the Sturm-Habicht sequence which generalized the work of W. Habicht [Comment. Math. Helv. 21, 99-116 (1948; Zbl 0029.24402)]. Although the above sequence is easier to compute compared with the conventional approach, such a sequence is not the Sturm sequence in the usual sense because there might be zero-polynomials in this sequence.

In this paper, the authors use a recursive method to construct GSS for two parametric polynomials via the subresultant polynomial chain of the two polynomials avoiding complicated division operations. They establish a connection between GSS and the subresultant polynomial chain of two polynomials and an algorithm that computes the leading coefficients and degrees of the GSS for the infinite interval case. The algorithm has been implemented in Maple and is efficient for constructing the leading terms of the GSS for polynomials with symbolic coefficients.

Reviewer: Manuel Ladra Gonzalez (Santiago)

### MSC:

12Y05 | Computational aspects of field theory and polynomials (MSC2010) |

12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |

13P99 | Computational aspects and applications of commutative rings |

### Keywords:

generalized Sturm sequence; subresultant polynomial; principal subresultant coefficient; polynomial remainder sequence; Maple### Software:

Maple
PDFBibTeX
XMLCite

\textit{H. Fu} et al., Sci. China, Ser. E 43, No. 1, 32--41 (2000; Zbl 0984.12008)

Full Text:
DOI

### References:

[1] | Tarski, A., A Decision Method for Elementary Algebra and Geometry, 2nd ed., Univ. of California Press, 1951. · Zbl 0044.25102 |

[2] | González-Vega, L., Lombardi, H., Recio, T. et al., Sturm-Habicht sequence, ISSAC-89 Proceedings, ACM Press, 1989, 136–146. |

[3] | González-Vega, L., Recio, T., Lombardi, H. et al., Sturm-Habicht sequence, determinants and real roots of univariate polynomials, Quantifier Elimination and Cylindrical Algebraic Decomposition (eds. Caviness, B. F., Johnson, J. R.), Berlin: Springer-Verlag, 1998, 300–316. · Zbl 0900.12002 |

[4] | Habicht, W., Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens, Comm. Math. Helvetici, 1948, 21: 99. · Zbl 0029.24402 · doi:10.1007/BF02568028 |

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[8] | Liang, S. X., Zhang, J. Z., A complete discrimination system for polynomials with complex coefficients and its automatic generation, Science in China, Ser. E, 1999, 42(2): 113. · Zbl 0949.12002 · doi:10.1007/BF02917106 |

[9] | Loos, R., Generalized polynomial remainder sequence, Computer Algebra: Symbolic and Algebraic Computation (ed. Buchberger, B.), Berlin: Springer-Verlag, 1983, 115–137. |

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