arXiv subjects

Tran Nam Son

Publications and source records attributed to Tran Nam Son.

9 recordsLinked to original sources

arXiv2026

Parametric decompositions into products of involutions

It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size $n$ of the matrices and the dimension $d$ of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed $n$ the optimal upper bound tends to infinity as $d\to \infty$. In some cases we are able to determine the optimal bound.

math.CV↗

arXiv2026

On the structural behavior of images of polynomials

The study of images of noncommutative polynomials on algebras has attracted considerable attention. We investigate polynomial images and the additive structures they generate in associative algebras, focusing on sums and products of values. Motivated by results on additive commutators, we show that finite sums of such products on a nonzero ideal must contains a nonzero ideal, with only minor exceptions. Consequently, for a simple algebra, the subring generated by the image of a noncentral polynomial coincides with the whole algebra, up to a small exceptional case. We further study representations of elements as sums of products of polynomial values, and examine products of additive commutators for matrices over division rings. To simplify multilinear polynomials, we introduce decomposable polynomials and show that, in many cases, their images equal the whole algebra. Finally, we consider polynomial commutators and prove that every noncommutative infinite simple algebra is generated by such elements, together with results on multiplicative commutators, including a complete description for real quaternions.

math.RA↗

arXiv2026

A nonlinear analogue of additive commutators

We study a nonlinear analogue of additive commutators, known as \textit{polynomial commutators}, defined by $p(ab) - p(ba)$ for a polynomial $p \in F[x]$ and elements $a, b$ in an algebra $R$ over a field $F$. Originally introduced by Laffey and West for matrices over fields, this notion is here extended to broader algebraic settings. We first show that in division rings, polynomial commutators can generate maximal subfields and even the entire ring as an algebra. In the matrix setting, we prove that matrices similar to ones with zero diagonal are polynomial commutators, and under mild assumptions, every matrix can be written as a product of at most three such commutators. Furthermore, we demonstrate that the matrix algebra can be decomposed as the sum of its center and the linear span of all polynomial commutators. Using the theory of rational identities in division rings, we also exhibit that the trace of a polynomial commutator in the matrix ring can be nonzero in noncommutative cases. Lastly, we explore the size of polynomial commutators via matrix norms.

math.RA↗

arXiv2025

On the injectivity of evaluation maps induced by polynomials on certain algebras

We explore the injectivity of the evaluation map eva f,A from Am A to A, where A is an associative algebra over a field F, and f is a polynomial in m \ge 1 variables with coefficients in F. Our investigation reveals that injectivity is possible only when m equal 1 and f has degree one; for functions in two or more variables, such injectivity is impossible.

math.RA↗

arXiv2025

A note on additive commutator groups in certain algebras

We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive commutators of $A$. Among our main considerations are the cases in which $A$ is the matrix ring over a division ring, a generalized quaternion algebra, or a semisimple finite-dimensional algebra. We also discuss some applications that do not necessarily require the decomposition, such as the case where $ A $ is the twisted group algebra of a locally finite group over a field of characteristic zero: if all additive commutators of $A$ are central, then $ A $ must be commutative.

math.RA↗

arXiv2025

On sums and products of diagonalizable matrices over division rings

This paper aims to continue the studies initiated by Botha in [Linear Algebra Appl. 273 (1998), 65-82; Linear Algebra Appl. 286 (1999), 37-44; Linear Algebra Appl. 315 (2000), 1-23] by extending them to matrices over noncommutative division rings. In particular, we show that every such matrix can be written as either a sum or a product of two diagonalizable matrices. The number $2$ is not valid under mild conditions on the center, similar to those in Botha's work on fields. By applying this result and other results obtained so far, we latter establish some Waring-type results for matrices.

math.RA↗

arXiv2025

Images of polynomial maps and the Ax-Grothendieck theorem over algebraically closed division rings

We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if $ f_1, \ldots, f_m $ are elements of the free associative algebra $ D\langle X_1, \ldots, X_m \rangle $ generated by $ m \geq 1 $ variables over an algebraically closed division ring $ D $ of finite dimension over its center $ F $, and if the induced map $ f = (f_1, \ldots, f_m) \colon D^m \to D^m $ is injective, then $ f $ must be surjective. With no condition on the dimension over the center, our second result is that $ p(D) = D $ if $ p $ is either an element in $ F\langle X_1, \ldots, X_m \rangle $ with zero constant term such that $ p(F) \neq \{0\} $, or a nonconstant polynomial in $F[x]$. Furthermore, we also establish some Waring type results. For instance, for any integer $ n > 1 $, we prove that every matrix in $ \mathrm{M}_n(D) $ can be expressed as a difference of pairs of multiplicative commutators of elements from $ p(\mathrm{M}_n(D)) $, provided again that $ D $ is finite-dimensional over $ F $.

math.RA↗

arXiv2023

Images of Multilinear Polynomials on Generalized Quaternion Algebras

The main goal of this paper is to extend [J. Algebra Appl. 20 (2021), 2150074] to generalized quaternion algebras, even when these algebras are not necessarily division rings. More precisely, in such cases, the image of a multilinear polynomial evaluated on a quaternion algebra is a vector space and we additionally provide a classification of possible images.

math.RA↗

arXiv2023

Products of Traceless and Semi-Traceless Matrices over Division Rings and their Applications

We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we establish the curious facts that every matrix over a division ring is a product of at most twelve traceless matrices as well as a product of at most four semi-traceless matrices. We also examine finitary matrices and certain images of non-commutative polynomials by applying the obtained so far results showing that the elements of some finite-dimensional algebras over a special field as well as that these of the matrix algebra over any division ring possess some rather interesting and non-trivial decompositions into products of at most four generalized commutators.

math.RA↗