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 J. Sib. Fed. Univ. Math. Phys., 2016, Volume 9, Issue 2, Pages 166–172 (Mi jsfu473)

On solvability of systems of symbolic polynomial equations

Oleg I. Egorushkin, Irina V. Kolbasina, Konstantin V. Safonov

Institute of Computer Science and Telecommunications, Reshetnev Siberian State Aerospace University, Krasnoyarsky Rabochy, 31, Krasnoyarsk, 660014, Russia

Abstract: Approaches to solving the systems of non-commutative polynomial equations in the form of formal power series (FPS) based on the relation with the corresponding commutative equations are developed. Every FPS is mapped to its commutative image — power series, which is obtained under the assumption that all symbols of the alphabet denote commutative variables assigned as values in the field of complex numbers. It is proved that if the initial non-commutative system of polynomial equations is consistent, then the system of equations being its commutative image is consistent. The converse is not true in general.
It is shown that in the case of a non-commutative ring the system of equations can have no solution, have a finite number of solutions, as well as having an infinite number of solutions, which is fundamentally different from the case of complex variables.

Keywords: non-commutative ring, polynomial equations, formal power series, commutative image.

DOI: https://doi.org/10.17516/1997-1397-2016-9-2-166-172

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UDC: 519.682+517.55
Accepted: 02.03.2016
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Citation: Oleg I. Egorushkin, Irina V. Kolbasina, Konstantin V. Safonov, “On solvability of systems of symbolic polynomial equations”, J. Sib. Fed. Univ. Math. Phys., 9:2 (2016), 166–172

Citation in format AMSBIB
\Bibitem{EgoKolSaf16} \by Oleg~I.~Egorushkin, Irina~V.~Kolbasina, Konstantin~V.~Safonov \paper On solvability of systems of symbolic polynomial equations \jour J. Sib. Fed. Univ. Math. Phys. \yr 2016 \vol 9 \issue 2 \pages 166--172 \mathnet{http://mi.mathnet.ru/jsfu473} \crossref{https://doi.org/10.17516/1997-1397-2016-9-2-166-172} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000412008200005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “O sovmestnosti sistem simvolnykh polinomialnykh uravnenii i ikh prilozhenii”, PDM. Prilozhenie, 2016, no. 9, 119–121
2. O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “Analog teoremy o neyavnom otobrazhenii dlya formalnykh grammatik”, PDM. Prilozhenie, 2017, no. 10, 149–151
3. O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “O primenenii mnogomernogo kompleksnogo analiza v teorii formalnykh yazykov i grammatik”, PDM, 2017, no. 37, 76–89
4. K. V. Safonov, O. I. Egorushkin, I. V. Kolbasina, “Sintaksicheskii analiz programm metodom integralnykh predstavlenii”, PDM. Prilozhenie, 2018, no. 11, 128–130
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