Parameterization of the discriminant set of a polynomial

Abstract

The discriminant set of a real polynomial is studied. It is shown that this set has a complex hierarchical structure and consists of algebraic varieties of various dimensions. A constructive algorithm for a polynomial parameterization of the discriminant set in the space of the coefficients of the polynomial is proposed. Each variety of a greter dimension can be geometrically considered as a tangent developable surface formed by one-dimensional linear varieties. The role of the directrix is played by the component of the discriminant set with the dimension by one less on which the original polynomial has a single multiple root and the other roots are simple. The relationship between the structure of the discriminant set and the partitioning of natural numbers is revealed. Various algorithms for the calculation of subdiscriminants of polynomials are also discussed. The basic algorithms described in this paper are implemented as a library for Maple.