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.
Received: 20.12.2015 Received in revised form: 24.01.2016 Accepted: 02.03.2016
Bibliographic databases:
Document Type:
Article
UDC:519.682+517.55
Language: English
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
\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}
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Linking options:
https://www.mathnet.ru/eng/jsfu473
https://www.mathnet.ru/eng/jsfu/v9/i2/p166
This publication is cited in the following 13 articles:
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “O reshenii lineinykh odnorodnykh grammatik, porozhdayuschikh lineinye yazyki”, PDM. Prilozhenie, 2024, no. 17, 123–125
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “O reshenii obschego algebraicheskogo uravneniya stepennymi ryadami i prilozhenii v teorii formalnykh grammatik”, PDM, 2023, no. 60, 106–113
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “Analog teoremy Kronekera — Kapelli dlya sistem nekommutativnykh lineinykh uravnenii, porozhdayuschikh lineinye yazyki”, PDM. Prilozhenie, 2023, no. 16, 124–126
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “O polinomialnykh grammatikakh, porozhdayuschikh beskonechnoe mnozhestvo yazykov”, PDM. Prilozhenie, 2022, no. 15, 78–80
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “O reshenii polinomialnykh grammatik i obschego algebraicheskogo uravneniya”, PDM. Prilozhenie, 2021, no. 14, 176–178
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “Geometricheskoe uslovie razreshimosti formalnykh grammatik”, PDM. Prilozhenie, 2020, no. 13, 106–108
V. V. Kishkan, K. V. Safonov, “Sintaksicheskii analiz monomov kontekstno-svobodnykh yazykov s uchetom poryadka primeneniya produktsii”, PDM. Prilozhenie, 2019, no. 12, 194–196
I. V. Kolbasina, K. V. Safonov, “Uslovie razreshimosti proizvolnykh formalnykh grammatik”, PDM. Prilozhenie, 2019, no. 12, 196–198
V V Kishkan, K V Safonov, R Yu Tsarev, “Syntactical analysis of context-free languages taking into account order of application of productions”, J. Phys.: Conf. Ser., 1333:3 (2019), 032072
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “Sintaksicheskii analiz programm metodom integralnykh predstavlenii”, PDM. Prilozhenie, 2018, no. 11, 128–130
O. I. Egorushkin, I. V. Kolbasina, K. V. Safonov, “Analog teoremy o neyavnom otobrazhenii dlya formalnykh grammatik”, PDM. Prilozhenie, 2017, no. 10, 149–151
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
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
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