Prospects for Using Finite Algebraic Rings for Constructing Discrete Coordinate Systems
Suleimenov I. Bakirov A.
March 2025Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2025#17Issue 3
The method of non-standard algebraic extensions based on the use of additional formal solutions of the reduced equations is extended to the case corresponding to three-dimensional space. This method differs from the classical one in that it leads to the formation of algebraic rings rather than fields. The proposed approach allows one to construct a discrete coordinate system in which the role of three basis vectors is played by idempotent elements of the ring obtained by a non-standard algebraic extension. This approach allows, among other things, the identification of the symmetry properties of objects defined through discrete Cartesian coordinates, which is important, for example, when using advanced methods of digital image processing. An explicit form of solutions of the equations is established that allow one to construct idempotent elements for Galois fields (Formula presented.) such that (Formula presented.) is divisible by three. The possibilities of practical use of the proposed approach are considered; in particular, it is shown that the use of discrete Cartesian coordinates mapped onto algebraic rings is of interest from the point of view of improving UAV swarm control algorithms.
algebraic rings , discrete Cartesian coordinates , galois fields , geometric problems , idempotent elements , information security , UAVs
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National Engineering Academy of the Republic of Kazakhstan, Almaty, 050010, Kazakhstan
Institute of Communication and Space Engineering, Gumarbek Daukeev Almaty University of Power Engineering and Communications, Almaty, 050013, Kazakhstan
National Engineering Academy of the Republic of Kazakhstan
Institute of Communication and Space Engineering
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