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Uspekhi Mat. Nauk, 2018, Volume 73, Issue 5(443), Pages 3–52 (Mi umn9838)  

This article is cited in 8 scientific papers (total in 8 papers)

Geometry of word equations in simple algebraic groups over special fields

N. L. Gordeevab, B. È. Kunyavskiĭc, E. B. Plotkinc

a Herzen State Pedagogical University, St. Petersburg, Russia
b St. Petersburg State University
c Bar-Ilan University, Ramat Gan, Israel

Abstract: This paper contains a survey of recent developments in the investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras, along with some new results on the images of word maps on algebraic groups defined over special fields: complex, real, $p$-adic (or close to such), or finite.
Bibliography: 174 titles.

Keywords: linear algebraic group, word map.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.661.2016/1.4
Israel Science Foundation 1623/16
Emmy Noether Mathematical Institute
Max-Planck-Institut für Mathematik
The research of the first author was financially supported by the Ministry of Science and Higher Education of the Russian Federation, project 1.661.2016/1.4. The research of the second and third authors was supported by ISF grant 1623/16 and the Emmy Noether Research Institute for Mathematics. The paper was written when the second author visited the MPIM (Bonn). The authors thank all these institutions.


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English version:
Russian Mathematical Surveys, 2018, 73:5, 753–796

Bibliographic databases:

UDC: 512.7
MSC: 20F70, 20G15
Received: 26.06.2018

Citation: N. L. Gordeev, B. È. Kunyavskiǐ, E. B. Plotkin, “Geometry of word equations in simple algebraic groups over special fields”, Uspekhi Mat. Nauk, 73:5(443) (2018), 3–52; Russian Math. Surveys, 73:5 (2018), 753–796

Citation in format AMSBIB
\by N.~L.~Gordeev, B.~\`E.~Kunyavski{\v\i}, E.~B.~Plotkin
\paper Geometry of word equations in simple algebraic groups over special fields
\jour Uspekhi Mat. Nauk
\yr 2018
\vol 73
\issue 5(443)
\pages 3--52
\jour Russian Math. Surveys
\yr 2018
\vol 73
\issue 5
\pages 753--796

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    This publication is cited in the following articles:
    1. Gnutov F., Gordeev N., “Recursive Sequences of Surjective Word Maps For the Algebraic Groups Pgl(2) and Sl2”, Arch. Math.  crossref  mathscinet  isi
    2. Klyachko A.A., Ryabtseva M.A., “The Dimension of Solution Sets to Systems of Equations in Algebraic Groups”, Isr. J. Math.  crossref  mathscinet  isi
    3. W. Cocke, M.-C. Ho, “Word maps in finite simple groups”, Arch. Math. (Basel), 113:6 (2019), 565–570  crossref  mathscinet  zmath  isi  scopus
    4. E. A. Egorchenkova, “Verbalnye otobrazheniya grupp Shevalle nad beskonechnymi polyami”, Voprosy teorii predstavlenii algebr i grupp. 34, Zap. nauchn. sem. POMI, 478, POMI, SPb., 2019, 108–127  mathnet
    5. F. A. Gnutov, N. L. Gordeev, “Ob obraze verbalnogo otobrazheniya s konstantami prostoi algebraicheskoi gruppy”, Voprosy teorii predstavlenii algebr i grupp. 34, Zap. nauchn. sem. POMI, 478, POMI, SPb., 2019, 78–99  mathnet
    6. Alexei  Kanel-Belov, Sergey Malev, Louis Rowen, Roman Yavich, “Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov–Kaplansky Conjecture”, SIGMA, 16 (2020), 071, 61 pp.  mathnet  crossref
    7. N. L. Gordeev, “O posledovatelnostyakh verbalnykh otobrazhenii kompaktnykh topologicheskikh grupp”, Voprosy teorii predstavlenii algebr i grupp. 35, Zap. nauchn. sem. POMI, 492, POMI, SPb., 2020, 94–124  mathnet
    8. R. V. Skuratovskii, “The derived subgroups of Sylow 2-subgroups of the alternating group, commutator width of wreath product of groups”, Mathematics, 8:4 (2020), 472  crossref  mathscinet  isi
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