Geodesic
Solvability of Independent Systems of Equations in Finitely Generated Nilpotent Groups
Matematičeskie zametki, Tome 110 (2021) no. 4, pp. 569-575

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that the solvability problem for a finite independent system of equations in a finitely generated nilpotent group can effectively be reduced to a similar problem in some finite quotient group of this group. Therefore, this problem is algorithmically solvable. This strengthens a theorem of A. G. Makanin on the residual finiteness and algorithmic solvability of a regular splittable equation in a finitely generated nilpotent group.
Mots-clés : Diophantine problem
Keywords: nilpotent group, regular equation, independent system, residual finiteness. 
V. A. Roman'kov. Solvability of Independent Systems of Equations in Finitely Generated Nilpotent Groups. Matematičeskie zametki, Tome 110 (2021) no. 4, pp. 569-575. http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a6/
@article{MZM_2021_110_4_a6,
     author = {V. A. Roman'kov},
     title = {Solvability of {Independent} {Systems} of {Equations} in {Finitely} {Generated} {Nilpotent} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {569--575},
     year = {2021},
     volume = {110},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a6/}
}
TY  - JOUR
AU  - V. A. Roman'kov
TI  - Solvability of Independent Systems of Equations in Finitely Generated Nilpotent Groups
JO  - Matematičeskie zametki
PY  - 2021
SP  - 569
EP  - 575
VL  - 110
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a6/
LA  - ru
ID  - MZM_2021_110_4_a6
ER  - 
%0 Journal Article
%A V. A. Roman'kov
%T Solvability of Independent Systems of Equations in Finitely Generated Nilpotent Groups
%J Matematičeskie zametki
%D 2021
%P 569-575
%V 110
%N 4
%U http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a6/
%G ru
%F MZM_2021_110_4_a6

[1] V. Roman'kov, “Equations over groups”, Groups Complex. Cryptol., 4:2 (2012), 191–239 | MR

[2] A. L. Shmelkin, “O polnykh nilpotentnykh gruppakh”, Algebra i logika, 6:2 (1967), 111–114 | MR

[3] A. G. Makanin, “O finitnoi approksimiruemosti uravnenii v konechno porozhdennykh nilpotentnykh gruppakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1992, no. 1, 48–51 | MR | Zbl

[4] A. I. Maltsev, “O gomomorfizmakh na konechnye gruppy”, Uch. zap. Ivanovskogo ped. in-ta, 18:5 (1958), 49–60

[5] J. C. C. McKinsey, “The decision problem for some classes of sentences whithout quantifiers”, J. Symbolic Logic, 8:3 (1943), 61–76 | DOI | MR

[6] Ph. Hall, Nilpotent Groups. Notes of Lectures Given at the Canadian Mathematical Congress Summer Seminar, University of Alberta, 12–30 August, 1957, Queen Mary College, London, 1969

[7] V. A. Romankov, “O nerazreshimosti problemy endomorfnoi svodimosti v svobodnykh nilpotentnykh gruppakh i v svobodnykh koltsakh”, Algebra i logika, 16:4 (1977), 457–471 | MR

[8] V. A. Romankov, “Ob uravneniyakh v svobodnykh metabelevykh gruppakh”, Sib. matem. zhurn., 20:3 (1979), 671–673 | MR | Zbl

[9] V. A. Roman'kov, Essays in Algebra and Cryptology: Solvable Groups, Dostoevsky Omsk State University Publishing House, Omsk, 2017

[10] V. A. Roman'kov, “Diophantine questions in the class of finitely generated nilpotent groups”, J. Group Theory, 19:3 (2016), 497–514 | MR

[11] N. N. Repin, “Uravneniya s odnoi neizvestnoi v nilpotentnykh gruppakh”, Matem. zametki, 34:2 (1983), 201–206 | MR | Zbl

[12] N. N. Repin, “Problema razreshimosti uravnenii s odnoi neizvestnoi v nilpotentnykh gruppakh”, Izv. AN SSSR. Ser. matem., 48:6 (1984), 1295–1313 | MR | Zbl