David Buzinski, Robin Winstanley
Let $K$ be an algebraically closed field of characteristic $0$ and let $M_n(K)$, $n \ge 3$, be the matrix ring over $K$. We will show that the image of any multilinear polynomial in four variables evaluated on $M_n(K)$ contains all matrices of trace $0$.
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The purpose of this paper is to describe the images of multilinear polynomials of arbitrary degree on the strictly upper triangular matrix algebra.
We describe the images of multilinear polynomials of degree up to four on the upper triangular matrix algebra.
We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3\times 3$ upper triangular matrix algebra over an infinite field.
Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices
本研究针对无限域上非零多重线性多项式在有限矩阵代数中的像问题，证明对任意正整数d存在s使其像含所有d阶迹零矩阵，进而得出其在全体有限矩阵代数中的像包含所有迹零有限矩阵的结论。
In this paper we prove that the image of multilinear polynomials evaluated on the algebra $UT_n(K)$ of $n\times n$ upper triangular matrices over an infinite field $K$ equals $J^r$, a power of its Jacobson ideal $J=J(UT_n(K))$. In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for $UT_n(K)$ is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree $r$ in terms of its coefficients. It turns out that the image of a multilinear…
On multilinear polynomial identities for $2\times2$ matrices in characteristic $2$
本文在特征2的无限域上，为$2\times2$矩阵代数$M_2(K)$找到了两个5次多线性恒等式，并与4次标准多项式共同生成所有次数不超过7的多线性恒等式，且证明三个生成元缺一不可。
A note on the image of graded multilinear polynomials on upper triangular matrices
研究上三角矩阵代数上分次变量多元多项式的像，证明在恒等分量交换时像总是子空间，并推广到逆极限和直极限。
Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak M}$ of every non-associative multilinear polynomial over ${\mathbb F}$ is a vector space. The same question we consider for two subalgebras: an algebra of the rock, the paper and the scissors without unit, and an algebra of trace zero elements with zero scalar part. Moreover in this paper we consider the questions of possible evaluations of homogeneous polyno…
Values of multilinear graded $*$-polynomials on upper triangular matrices of small dimension
本文研究多重线性分次*-多项式在小维数上三角矩阵上的像，证明其在 UT_2 及非平凡分次 UT_3 上为齐次向量空间，并说明该结论最优，从而表明 L'vov-Kaplansky 猜想在带分次对合的代数中不成立。
In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear pol…
In this paper we prove the generalized Kaplansky conjecture for the Jordan algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of multilinear polynomial must be either $\{0\}$, $\mathbb{R}$, the space of pure elements $V$, or $J_n$.
In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of …
Upper triangular matrices with superinvolution: identities and images of multilinear polynomials
本文完整刻画了UT_4(F)上由序列(0,1,0,1)诱导分次及超辛超级对合的恒等式，并证明UT_n(F)上多重线性多项式像为向量空间当且仅当n≤3，推进了L'vov-Kaplansky猜想的类比研究。
A celebrated theorem of Shoda states that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or, equivalently, that the set of values of the polynomial f(x,y)=xy-yx on the nxn-matrix K-algebra contains all matrices with trace 0. We generalize Shoda's theorem by showing that every nonzero multilinear polynomial of degree at most 3, with coefficients in K, has this property. We further conjecture that this holds for every nonzero multilinear polynomial with coefficients in K of degree m, provided that m is at most n+1.
Given a positive integer d, the Kaplansky-Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f on the matrix algebra M_d(C) is a vector subspace. In this article the technique of using one-wiggle families of Sylvester's clock-and-shift matrices is championed to establish the conjecture for polynomials f of degree three when d is even or d<17.
We delineate the image of multilinear Lie polynomial of degree $2$ evaluated on $L$ where $L$ is a finite-dimensional nilpotent Lie algebra over field $k$ with $\dim L' \leq 4$.
A short note on the image of multilinear graded polynomials on matrix algebras
本文研究典范$C_n$分次矩阵代数上多重线性分次多项式的像子空间，证明分次平凡分量的每个$C_n$子模均为对应多项式的像，否定了de Castilho与Centrone2023年的猜想。
Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper we settle the analogous conjecture for a quaternion algebra.
In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over Bézout domains as to when such a matrix is a product of idempotent matrices.
Images of Multilinear Polynomials on Generalized Quaternion Algebras
本文推广了多重线性多项式在广义四元数代数上的像的研究，证明其像为向量空间并给出分类，即使代数非可除环。
Any multilinear non-central polynomial $p$ (in several noncommuting variables) takes on values of degree $n$ in the matrix algebra $M_n(F)$ over an infinite field $F$. The polynomial $p$ is called {\it $ν$-central} for $M_n(F)$ if $p^ν$ takes on only scalar values, with $k$ minimal such. Multilinear $ν$-central polynomials do not exist for any $ν$ with $n>3$, thereby answering a question of Drensky. Saltman proved that an arbitrary polynomial $p$ cannot be $ν$-central for $M_n(F)$ for $n$ odd unless $n$ is prime; we show for $n$ even, that $ν$ must be 2.
Von Staudt Constructions for Skew-Linear and Multilinear Matroids
本文利用von Staudt构造将斜线性与多重线性拟阵的表示问题转化为代数问题，证明了除环上多类拟阵表示问题的不可判定性，并给出两类拟阵性质差异的具体示例。
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 $M_n(K)$ over a field $K$ endowed with its canonical $\mathbb{Z}_n$-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field $\mathbb Q$ of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on $n\times n$ matrices, where $n$ is a prime number. We confirm such conjecture for polynomials of degree …
We present new additive results for the group invertibility in a ring. Then we apply our results to block operator matrices over Banach spaces and derive the existence of group inverses of $2\times 2$ block operator matrices. These generalize many known results, e.g., Benitez, Liu and Zhu(Linear Multilinear Algebra, {\bf 59}(2011), 279--289) and Zhou, Chen and Zhu(Comm. Algebra, {\bf 48}(2020),676-690).
The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization
本文针对模为1的四元数$\phi$，研究四元数矩阵的$\phi$-共轭相关性质，并将Autonne-Takagi分解推广至所有单位四元数，肯定回答了Horn与Zhang提出的问题。
Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices over the complex numbers and view it as a crossed product with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the regular representation and embedding $S_k$ in $M_k(\mathbb{C})$ in the usual way. This induces a natural $G$-grading on $M_k(\mathbb{C})$ which we call a crossed-product grading. We study the graded $*$-identities for $M_k(\mathbb{C})$ equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra wit…
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 ${\rm sl}_n(K)$ 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 $n\geq 3$ i…
Polynomials that preserve nonnegative matrices
本文研究保持固定阶非负矩阵的多项式集合 $\mathscr{P}_n$，针对其系数给出新结果，推广偶部与奇部并证明与已有构造等价，为后续研究提供方向。
Let $G:=SO(2n)$ be the even special orthogonal group over $\mathbb{C}$ and let $M_{2n}^+$ (resp. $M_{2n}^-$) be the space of symmetric (resp. skew-symmetric) complex matrices with respect to the usual transposition. We study the structure of the space $B^+:=\left(\bigwedge (M_{2n}^{+})^*\otimes M_{2n}^-\right)^G$, the space of $G-$equivariant skew-symmetric matrix valued alternating multilinear maps on the space of symmetric $n-$tuples of matrices, with $G$ acting by conjugation. We prove that $B^+$ is a free module over a certain subalgebra of invariants $A:=\left(\bigwedge (M_{2n}^{+})^*\rig…
Properties of the cone of polynomials of fixed degree that preserve nonnegative matrices
本文研究保持非负矩阵不变的多项式锥的几何性质，证明高次多项式生成的锥非多面体，回答了Loewy关于中心项负值程度的问题，并给出该锥在单位球面上体积的初步界。
We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of $\mathbb ZS_n$-submodules in the $\mathbb ZS_n$-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding $\mathbb ZS_n$-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper tr…
The well-known Lvov-Kaplansky conjecture states that the image of a multilinear polynomial $f$ evaluated on $n\times n$ matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if $m=deg( f)$ and $n\geq m-1$ then its image contains the set of trace zero matrices. Such conjecture has been proved for polynomials of degree $m \leq 4$. 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 \g…
Makar-Limanov's problem on values of polynomials on matrices
该研究通过构造特定矩阵，证明了非交换多项式在矩阵上的值可构造上三角矩阵，并探讨了其特征值的代数无关性。
Images of polynomials with involution on $2\times 2$ matrices
研究域上 $2\times 2$ 矩阵代数中多重线性 $*$-多项式的像，对于转置对合和辛对合，证明了像要么是向量子空间，要么包含基，并补充了 Brešar 和 Klep 关于有限维中心单代数上 $*$-多项式像的线性张成的定理。
Let $f$ be a polynomial in the free algebra over a field $K$, and let $A$ be a $K$-algebra. We denote by $§_A(f)$, $\A_A(f)$ and $\I_A(f)$, respectively, the `verbal' subspace, subalgebra, and ideal, in $A$, generated by the set of all $f$-values in $A$. We begin by studying the following problem: if $§_A(f)$ is finite-dimensional, is it true that $\A_A(f)$ and $\I_A(f)$ are also finite-dimensional? We then consider the dual to this problem for `marginal' subspaces that are finite-codimensional in $A$. If $f$ is multilinear, the marginal subspace, $\widehat§_A(f)$, of $f$ in $A$ is the set of …
Convergence for noncommutative rational functions evaluated in random matrices
本文将自由概率中非交换多项式在随机矩阵模型下的渐近收敛性推广至非交换有理函数，回答了 Roland Speicher 的公开问题，并提供了一种逼近自由场的非交换概率方法。
A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.
Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from linear algebra that offers elegant solutions to all these questions. The doodles, known as trace diagrams, are graphs labeled by matrices which have a correspondence to multilinear functions. This correspondence permits computations in linear algebra to be performed using diagrams. The result is an elegant theory from which standard constructions of linear al…
The nearest circulant approximation of a real Toeplitz matrix in the Frobenius norm is derived. This matrix is symmetric. It is proven that symmetric circulant matrices are the only real circulant matrices with all real eigenvalues. The Frobenius norm of the difference between this approximation and the Toeplitz matrix for the case of a Toeplitz matrix displaying exponential decay is evaluated using an expression of $\sum_{n = 1}^N n^k p^n$ in terms of the first $k$ geometric moments. Compared to a classic approximation the nearest circulant displays dramatically better behaviour in any finite…
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