The trace algebra C_{nd} is generated by all traces of products of d generic n x n matrices. Minimal generating sets of C_{nd} and their defining relations are known for n < 3 and n = 3, d=2. This paper states a minimal generating set and their defining relations for n=d=3. Furthermore the computations yield a description of C_{33} as a free module over the ring generated by a homogeneous system of parameters.
Given any commutative ring $R$, a commutator of two $n\times n$ matrices over $R$ has trace $0$. In this paper, we study the converse: whether every $n \times n$ trace $0$ matrix is a commutator. We show that if $R$ is a Bézout domain with algebraically closed quotient field, then every $n\times n$ trace $0$ matrix is a commutator. We also show that if $R$ is a regular ring with large enough Krull dimension relative to $n$, then there exist a $n\times n$ trace $0$ matrix that is not a commutator. This improves on a result of Lissner by increasing the size of the matrix allowed for a fixed $R$.…
On pairs of matrices generating matrix rings and their presentations
设$M_n(\mathbb{Z})$为整数元$n$乘$n$矩阵环，$n \geq 2$ .本文研究了生成$M_n(\mathbb{Z})$为环的对$(A,B) \in M_n(\mathbb{Z})^2$的集合$G_n(\mathbb{Z})$.我们使用几个演示的$M_{n}(\mathbb{Z})$与发电机$X=\sum_{i=1}^n E_{i+1,i}$和$Y=E_{11}$，以获得以下结果。\begin{enumerate} \item Let $k \geq 1$. Then the rings $M_n(\mathbb{Q})^k$ and $\bigoplus_{j=1}^{k} M_{n_j} (\mathbb{Z})$, where $n_1, ..., n_k \geq 2$ are pairwise relatively prime, have presentations with 2 generators and finitely many relations. \item Let $D$ be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 rel…
A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0, \ldots, A_r)$. We will refer to $A$ as a presentation of $H$. Another presentation $B = (B_0, B_1, \ldots, B_r)$ of $H$ can be obtained by choosing $g_1, g_2 \in \operatorname{GL}_n$ and setting $B_i = g_1 A_i g_2$ for every $i = 0, 1, \ldots, r$. In this case $A$ and $B$ are called equivalent. The second author and A. Vistoli have shown that for $r \geq 3$ a gen…
In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynom…
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.
Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $ψ:V\to V$ that satisfy $ψ(A^k)=ψ(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $ϕ,ψ:V\to V$ that satisfy "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}…
