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Spécialisation de la suite de Sturm et sous-résultants. I. (Specialization of the Sturm sequence and subresultants. I). (French) Zbl 0732.68059
This paper is the first one of a series of two devoted to the specialization of the Sturm sequence and the subresultants. Sturm sequences allow to compute the number of roots of a polynomial and generalized Sturm sequences to compute the number of real roots of a polynomial verifying a given polynomial inequality. It is desirable in computer algebra to design improvements of Sturm sequences for which it is possible to control the growth of the computation and to avoid denominators in order to be able to specialize the coefficients.
The first part contains the definition and properties of the generalized Sturm sequence and explains why its behaviour under specialization is bad.
The second part contains a developed, precise and detailed analysis of the theory of subresultants (correcting some errors in the literature). The subresultants are polynomials proportional to the polynomials in the sequence of the remainders of the Euclidean algorithm. Their coefficients are defined in determinants, hence their computation can be done with a good control of the coefficients growth in indeterminate computations and they enjoy a good behaviour under specialization. Precise algorithms in order to compute them efficiently are described and discussed.
Reviewer: L.González-Vega

68W30 Symbolic computation and algebraic computation
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
68Q25 Analysis of algorithms and problem complexity
Full Text: DOI EuDML
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