The images of multilinear polynomials on strictly upper triangular matrices

Fagundes, Pedro S.

doi:10.1016/j.laa.2018.11.014

Cited by 21 publications

(7 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(Analogously, although the finite basis problem for multilinear identities is not yet settled in nonzero characteristic, there are counterexamples for completely homogeneous polynomials, cf. [ 39. Note according to Lemma 1.38 that if K has infinite transcendence degree over F we can take as many generic elements as we need.…”

Section: Generic Matricesmentioning

confidence: 99%

“…In [39] Fagundes denotes by U T (k) n for k ≥ 0 the set of strictly upper triangular matrices which, besides the main diagonal, also have k zero diagonals located above the main diagonal, and proves that if p is a multilinear polynomial evaluated on U T (0) n of degree m then its image is either {0} or U T (m−1) n . In particular, Im p is a vector space.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Kanel-Belov¹,

Malev

Rowen

et al. 2020

SIGMA

View full text Add to dashboard Cite

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.

show abstract

Section: Generic Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Kanel-Belov¹,

Malev

Rowen

et al. 2020

SIGMA

View full text Add to dashboard Cite

show abstract

“…Some generalizations of the L'vov-Kaplansky Conjecture have been studied considering algebras other than M n (K). The possible images of a multilinear polynomial are known for the algebra of upper triangular matrices U T n (K) and for its subalgebra of strictly upper triangular matrices [9,17,8] and also for the algebra of quaternions [19] and for some classes of simple Jordan algebras [20].…”

Section: Introductionmentioning

confidence: 99%

Images of graded polynomials on matrix algebras

Centrone¹,

Mello²

2022

Preprint

View full text Add to dashboard Cite

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra Mn(K) over a field K endowed with its canonical Zn-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field Q of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on n × n matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over Mn(K) when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on M 2 (K).

show abstract

“…This conjecture motivated other studies related to images of polynomials. For instance, papers on images of polynomials on some subalgebras of M n (K), images of Lie, and Jordan polynomials on Lie and Jordan algebras have been published since then (see [3,7,8,9,12,15,16]). For a nice compilation of results on images of polynomials, we recommend the survey [13].…”

Section: Introductionmentioning

confidence: 99%

The Mesyan Conjecture: a restatement and a correction

Fagundes,

de Mello,

Santos

2021

Preprint

View full text Add to dashboard Cite

The well-known Lvov-Kaplansky conjecture states that the image of a multilinear polynomial f evaluated on n × n matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if m = deg f and n ≥ m − 1 then its image contains the set of trace zero matrices. Such conjecture has been proved for polynomials of degree m ≤ 4. The proof of the case m = 4 contains an error in one of the lemmas. In this paper, we correct the proof of such lemma and present some evidences which allow us to state the Mesyan conjecture for the new bound n ≥ m+1 2 , which cannot be improved.

show abstract

The images of multilinear polynomials on strictly upper triangular matrices

Abstract: The purpose of this paper is to describe the images of multilinear polynomials of arbitrary degree on the strictly upper triangular matrix algebra.

Cited by 21 publications

References 14 publications

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Images of graded polynomials on matrix algebras

The Mesyan Conjecture: a restatement and a correction

Contact Info

Product

Resources

About