Structure of discriminant set of real polynomial

A. B. Batkhin

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 23-34.

Résumé

The problem of description the structure of the discriminant set of a real polynomial often occurs in solving various applied problems, for example, for describing a set of stability of stationary points of multiparameter systems, for computing the normal form of a Hamiltonian system in vicinity of equilibrium in the case of multiple frequencies. This paper considers the structure of the discriminant set of a polynomial with real coefficients. There are two approaches to its study. The first approach is based on the study of zeroes of ideals formed by the set of subdiscriminants of the original polynomial. Different ways of computing subdiscriminants are given. There is proposed to investigate the singular points of the discriminant set in the second approach. By the methods of computer algebra it is shown that for small values of the degree of the original polynomial, both approaches are equivalent, but the first one is preferred because of smaller ideals. Proposed constructive algorithm for obtaining polynomial parameterization of the discriminant set in the space of coefficients of the polynomial. From the applied point of view the most interesting is the description of the components of codimension 1 of the discriminant set. It is this component divides the space of the coefficients into the domains with the same structure of the roots of the polynomial. The set of components of different dimensions of the discriminant set has a hierarchical structure. Each component of higher dimensions can be considered as some kind of tangent developable surface which is formed by linear varieties of respective dimension. The role of directrix of this component performs a variety of dimension one less than that on which the original polynomial has only multiple zero and the remaining zeroes are simple. Starting with a one-dimensional algebraic variety of dimension 1 on which the original polynomial has the unique zero of maximal multiplicity, in the next step of the algorithm we obtain the description of the variety on which the polynomial has a pair of zeroes: one simple and another multiple. Repeating sequentially the steps of the algorithm, the resulting parametric representation of components of codimension 1 of the discriminant set can be obtained. Examples of the discriminant set of a cubic and quartic polynomials are considered. Bibliography: 15 titles.

Détail

Keywords: discriminant set, singular point, rational parametrization.

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