Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
Let be a polynomial in several non-commuting variables with coefficients in a field of arbitrary characteristic. It has been conjectured that for any , for multilinear, the image of evaluated on the set Mₙ() of by matrices is either zero, or the set of scalar matrices, or the set slₙ() of matr...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210777 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture. Alexei Kanel-Belov, Sergey Malev, Louis Rowen and Roman Yavich. SIGMA 16 (2020), 071, 61 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862734138505691136 |
|---|---|
| author | Kanel-Belov, Alexei Malev, Sergey Rowen, Louis Yavich, Roman |
| author_facet | Kanel-Belov, Alexei Malev, Sergey Rowen, Louis Yavich, Roman |
| citation_txt | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture. Alexei Kanel-Belov, Sergey Malev, Louis Rowen and Roman Yavich. SIGMA 16 (2020), 071, 61 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let be a polynomial in several non-commuting variables with coefficients in a field of arbitrary characteristic. It has been conjectured that for any , for multilinear, the image of evaluated on the set Mₙ() of by matrices is either zero, or the set of scalar matrices, or the set slₙ() of matrices of trace 0, or all of Mₙ(). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for = 2 in Section 2, some decisive results for = 3 in Section 3, and partial information for ≥ 3 in Section 4, also for non-multilinear polynomials. In addition, we consider the case of not algebraically closed, and polynomials evaluated on other finite-dimensional simple algebras (in particular, the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other research, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of the art, and on the other hand, provide counterexamples showing the boundaries of generalizations.
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| first_indexed | 2026-04-17T16:01:46Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210777 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T16:01:46Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kanel-Belov, Alexei Malev, Sergey Rowen, Louis Yavich, Roman 2025-12-17T14:36:56Z 2020 Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture. Alexei Kanel-Belov, Sergey Malev, Louis Rowen and Roman Yavich. SIGMA 16 (2020), 071, 61 pages 1815-0659 2020 Mathematics Subject Classification: 16H05;16H99;16K20;16R30;16R40;17B99 arXiv:1909.07785 https://nasplib.isofts.kiev.ua/handle/123456789/210777 https://doi.org/10.3842/SIGMA.2020.071 Let be a polynomial in several non-commuting variables with coefficients in a field of arbitrary characteristic. It has been conjectured that for any , for multilinear, the image of evaluated on the set Mₙ() of by matrices is either zero, or the set of scalar matrices, or the set slₙ() of matrices of trace 0, or all of Mₙ(). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for = 2 in Section 2, some decisive results for = 3 in Section 3, and partial information for ≥ 3 in Section 4, also for non-multilinear polynomials. In addition, we consider the case of not algebraically closed, and polynomials evaluated on other finite-dimensional simple algebras (in particular, the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other research, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of the art, and on the other hand, provide counterexamples showing the boundaries of generalizations. We would like to thank M. Bresar and B. Kunyavskii, and express our special gratitude to E. Plotkin for interesting and fruitful discussions. We would also like to thank the anonymous referees whose suggestions contributed significantly to the quality of the paper. The second and third-named authors were supported by the ISF (Israel Science Foundation) grant 1994/20. The first-named author was supported by the Russian Science Foundation grant No. 17-11-01377. The second and fourth-named authors were supported by the Israel Innovation Authority, grant no. 63412: Development of A.I. based platform for e-commerce. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture Article published earlier |
| spellingShingle | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture Kanel-Belov, Alexei Malev, Sergey Rowen, Louis Yavich, Roman |
| title | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture |
| title_full | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture |
| title_fullStr | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture |
| title_full_unstemmed | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture |
| title_short | Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture |
| title_sort | evaluations of noncommutative polynomials on algebras: methods and problems, and the l'vov-kaplansky conjecture |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210777 |
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