Vol 44, No 2 (2018)

A generalization of the classical discriminant of the real polynomial defined using the linear Hahn operator that decreases the degree of the polynomial by one is studied. The structure of the generalized discriminant set of the real polynomial, i.e., the set of values of the polynomial coefficients at which the polynomial and its Hahn operator image have a common root, is investigated. The structure of the generalized discriminant of the polynomial of degree n is described in terms of the partitions of n Algorithms for the construction of a polynomial parameterization of the generalized discriminant set in the space of the polynomial coefficients are proposed. The main steps of these algorithms are implemented in a Maple library. Examples of calculating the discriminant set are discussed.

With the help of computer algebra methods properties of the algebraic system that determines stationary motions of an axisymmetric satellite moving along a circular orbit subject to gravitational and active damping moments are investigated. The main attention is paid to the study of the conditions for the existence of stationary satellite motions. Computer algebra methods based on the algorithm for the construction of Gröbner bases are applied to reduce the satellite stationary motion system of six algebraic equations in six variables to a single algebraic equation in one variable that determines all stationary motions of the satellite. A classification of domains with equal numbers of stationary solutions is carried out using algebraic methods for constructing discriminant hypersurfaces. The effectiveness of various algorithms for constructing Gröbner bases for the solution of the problem under consideration was compared.

In this paper, the concept of satellite unknowns in differential systems with selected unknowns is considered in the context of irreducible systems. It is proved that any unselected unknown in an irreducible differential system is linearly satellite for any nonempty set of selected unknowns. An algorithm for factorization of differential systems is proposed that is not always applicable but executable in polynomial time. Cases where the algorithm cannot be applied are also recognized in polynomial time.

Systems of linear q-difference equations with polynomial coefficients are considered. Equations in the system may have arbitrary orders. For such systems, algorithms for searching polynomial, rational, and hypergeometric solutions, as well as solutions in the form of Laurent series, are suggested. Implementations of these algorithms are discussed.