Vol 42, No 2 (2016)

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.

As was shown earlier, for a linear differential–algebraic system A₁y′ + A₀y = 0 with a selected part of unknowns (entries of a column vector y), it is possible to construct a differential system ỹ′ = Bỹ, where the column vector ỹ is formed by some entries of y, and a linear algebraic system by means of which the selected entries that are not contained in ỹ can be expressed in terms of the selected entries included in ỹ. In the paper, sizes of the differential and algebraic systems obtained are studied. Conditions are established under the fulfillment of which the size of the algebraic system is determined unambiguously and the size of the differential system is minimal.

This paper presents a method and a corresponding algorithm for constructing volume forms (and related forms that act as kernels of integral representations) on toric varieties from a convex integer polytope. The algorithm is implemented in the Maple computer algebra system. The constructed volume forms are similar to the volume forms of the Fubini–Study metric on a complex projective space and can be used for constructing integral representations of holomorphic functions in polycircular regions of a multidimensional complex space.

Darboux transformations of type I are invertible Darboux transformations with explicit short formulas for inverse transformations. These transformations are invariant with respect to gauge transformations, and, for gauge transformations acting on third-order hyperbolic operators of two variables, a general-form system of generating differential invariants is known. In the paper, first-order Darboux transformations of type I for this class of operators are considered. The corresponding operator orbits are directed graphs with at most three edges originating from each vertex. In the paper, an algorithm for constructing such orbits is suggested. We have derived criteria for existence of first-order Darboux transformations of type I in terms of the generating invariants, formulas for transforming invariants, and the so-called “triangle rule” property of orbits. The corresponding implementation in the LPDO package is described. The orbits are constructed in two different forms, one of which outputs the graph in the format of the well-known built-in Maple package Graph Theory.