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Finding the inverse of a polynomial modulo in the ring Z[x] based on the method of undetermined coefficients

Yakymenko Ihor, Kasianchuk Mykhailo, Karpinski Mikolaj, Shevchuk Ruslan, Shylinska Inna

Computer Science

2024

T. 25 (2)

239--252

This paper presents the theoretical foundations of finding the inverse of a polynomial modulo in the ring Z[x] based on the method of undetermined coefficients. The use of the latter makes it possible to significantly reduce the time complexity of calculations avoiding the operation of finding the greatest common divisor. An example of calculating the inverse of a polynomial modulo inthe ring Z[x] based on the proposed approach is given. Analytical expressions of the time complexities of the developed and classical methods depending on the degrees of polynomials are built. The graphic dependence of the complexity of performing the operation of finding the inverse of a polynomial in the ring Z[x] is presented, which shows the advantages of the method based on undetermined coefficients. It is found that the efficiency of the developed method increases logarithmically with an increase in the degrees of polynomials.

Combined algorithm for finding conservation laws and implectic operators for the Boussinesq-Burgers nonlinear dynamical system and its finite dimensional reductions

Kindybaliuk A., Prytula M.

Journal of Applied Mathematics and Computational Mechanics

2014

Vol. 13, nr 3

85--99

In the article the combined algorithm for finding conservation laws and implectic operators has been proposed. Using the Novikov-Bogoyavlensky method the finite dimensional reductions have been found. The structure of invariant submanifolds has been examined. Having analyzed phase portraits of Hamiltonian systems, partial periodical solutions have been found.