## Abstract

Let p be a polynomial in several non-commuting variables with coefficients in a field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set M_{n}(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl_{n}(K) of matrices of trace 0, or all of M_{n}(K). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for n = 2 in Section 2, some decisive results for n = 3 in Section 3, and partial information for n ≥ 3 in Section 4, also for non-multilinear polynomials. In addition we consider the case of K 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 researches, 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 art, and in the other hand providing counterexamples showing the boundaries of generalizations.

Original language | English |
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Article number | 071 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 16 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© 2020, Institute of Mathematics. All rights reserved.

### Funding

We would like to thank M. Breˇsar, 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 level 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 Israel Innovation Authority, grant no. 63412: Development of A.I. based platform for e commerce.

Funders | Funder number |
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Israel Innovation Authority | 63412 |

Israel Science Foundation | 1994/20 |

Russian Science Foundation | 17-11-01377 |

## Keywords

- L’vov–Kaplansky conjecture
- Multilinear polynomial evaluations
- Noncommutative polynomials
- PI algebras
- Power central polynomials
- The Deligne trick