Vol 45, No 2 (2019)

Methods for computing the generalized discriminant set of a polynomial the roots of which satisfy a linear relation are considered. Using a q-analog of the classical elimination theory and computer algebra algorithms, methods for computing the parametric representation of this set are described, and these methods are implemented in a Maple library. The operation of these methods is demonstrated by an example.

Computer algebra methods are used to investigate properties of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies connected by a spherical hinge that moves along a circular orbit under the action of gravitational torque. To determine the equilibrium orientations for the system of two bodies, the system of 12 stationary algebraic equations is decomposed using linear algebra methods and algorithms for Gröbner basis construction. Depending on the parameters of the problem, the number of equilibria is found by analyzing the real roots of the algebraic equations from the Gröbner basis constructed. Evolution of the conditions for equilibria existence in the dimensionless parameter space of the problem is investigated. The effectiveness of the algorithms for Gröbner basis construction is analyzed depending on the number of parameters for the problem under consideration.

In the Maple computer algebra system, a set of recurrence relations and associated generating functions is derived for the number of near-perfect matchings on \({{C}_{m}} \times {{C}_{n}}\) tori of odd order at fixed values of the parameter m (\(3 \leqslant m \leqslant 11\)). The identity of the recurrence relations for the number of perfect and near-perfect matchings is revealed for the same value of m. An estimate for the number of near-perfect matchings is obtained at large odd m when \(n \to \infty \).

Computer algebra methods are widely employed in various branches of mathematics, physics, and other sciences. Simplification of algebraic expressions with indices is one of the important problems. Tensor expressions are the most typical example of these expressions. This paper briefly describes some basic methods for reducing expressions with indices to canonical form. The focus is placed on taking into account the properties of symmetries with respect to various permutations of indices in elementary symbols, symmetries associated with renaming summation indices, and general linear relationships among them. This paper also gives a definition of canonical representation for polynomial (multiplicative) expressions of variables with abstract indices that results from averaging the initial expression over the action of some finite group (signature stabilizer). In practice, e.g., for expressions of Riemann curvature tensors, the proposed algorithms show high efficiency.