No 2 (2025)

Annual report on the work of the research seminar on computer algebra.

A heuristic method that allows us to determine in advance the cases of integrability of autonomous dynamical systems with a polynomial right-hand side is used. The capabilities of this method are demonstrated using examples of two- and three-dimensional dynamical systems with a quadratic nonlinearity. A significant achievement compared to previous works is the ability to study systems of a general type without resonances in the linear parts, which is achieved by generalizing the results of resonance cases. Thus, it becomes possible to use the obtained results when working with dynamical models of real systems.

Systems of nonalgebraic equations containing entire functions of several complex variables are considered. The number of real zeros of such systems is studied using computer algebra methods. For this purpose, a computer implementation of Newton’s recurrences and formulas for the resultant of the functions under study in Maple is proposed. The relevance of this problem is due to the fact that in applied problems, e.g., in the equations of chemical kinetics, it is necessary to determine the number of stationary states of the system.

Systems of form y(x)′ = A(x)y(x) are considered, with matrix elements being initial segments of unknown infinite power series. The concept of a cyclic vector is generalized to the case of these systems by introducing the concept of a strongly cyclic vector. A method for checking the strong cyclicity of a vector is discussed. A sufficient condition that allows one to check the strong cyclicity of a vector by the form of the matrix of derivatives with respect to the system is derived.

We consider the problem of finding a binary solution to a system of linear equations modulo three. In case the number of equations is less than a sufficiently slowly growing function of the number of variables, a new polynomial-time algorithm is proposed to recognize the existence of a binary solution to such a system. The algorithm is based on the note that if the coefficient matrix contains non-zero columns proportional to each other, then the elimination of the corresponding variables preserves the property of having no binary solution to the system. In particular, every system of two equations in five variables allows the elimination of some variables that preserves the property of having no binary solution to the system. Based on these results, we propose an errorless heuristic algorithm, which is implemented using the Python programming language. The NumPy library is used to represent matrices and perform basic operations. The input is the augmented matrix. An empirical running time estimate has been calculated using the implementation. It has been experimentally shown that the algorithm is more efficient for sparse systems of equations. Obviously, the binary search method allows finding a binary solution to the system when one exists. This observation opens up the possibility of practical use, in particular, for solving problems of mathematical biology.