Title | Implementation of the algorithm for finding Groebner bases of polynomial rings ideals from several variables in C++ |
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Authors | A. A. Pivkin^{1}, L. A. Sukharev^{1}^{1}National Research Ogarev Mordovia State University |

Annotation | The article is devoted to the implementation of the Buchberger algorithm for constructing the Groebner basis of the polynomial ring ideal of several variables over a field of complex numbers in the C++ programming language. Hilbert’s wellknown theorem on the basis of the polynomials ring ideal in several variables ensures the existence of a finite basis for such an ideal. The well-known Buchberger algorithm constructs the Groebner basis of this ideal In the case of specifying an ideal by a finite set of polynomials. The paper describes in detail the computer implementation of one version of the algorithm under the condition of lexicographic ordering of polynomials. In particular, the following are considered: the representation of polynomials from several variables with complex coefficients in a computer, the assignment of arithmetic operations on them, the study of the meshing of two basis polynomials, the implementation of the reduction operation and the minimization of the Grobner basis. Examples of the program’s operation and comments on the main points of the program code are given. |

Keywords | Gröbner basis, Buchberger’s algorithm. |

Citation | Pivkin A. A., Sukharev L. A. ''Implementation of the algorithm for finding Groebner bases of polynomial rings ideals from several variables in C++'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 14-18, 2022). Saransk: SVMO Publ, 2022. - pp. 160-167. Available at: https://conf.svmo.ru/files/2022/papers/paper26.pdf. - Date of access: 30.11.2022. |

**© SVMO, National Research Mordovia State University, 2022**

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