相关论文: An improved bound for the length of matrix algebra…

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq…

Let $\mathcal A$ be an $\mathbb F$-algebra and let $\mathcal S$ be its generating set. The length of $\mathcal S$ is the smallest number $k$ such that $\mathcal A$ equals the $\mathbb F$-linear span of all products of length at most $k$ of…

We prove an exponential upper bound for the number $f(m,n)$ of all maximal triangulations of the $m\times n$ grid: \[ f(m,n) < 2^{3mn}. \] In particular, this improves a result of S. Yu. Orevkov (1999).

If $\A$ is an algebra over a field $\F$ and $\s$ is a generating set of $\A$, the length of $\s$ indicates the maximal length needed to express an arbitrary element of $\A$ as a linear combination of words in the elements of $\s$. The…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest…

We prove two results about width of words in $SL_n(\mathbb{Z})$. The first is that, for every $n \geq 3$, there is a constant $C(n)$ such that the width of any word in $SL_n(\mathbb{Z})$ is less than $C(n)$. The second result is that, for…

We show that the number of all maximal $\alpha$-gapped repeats and palindromes of a word of length $n$ is at most $3(\pi^2/6 + 5/2) \alpha n$ and $7 (\pi^2 / 6 + 1/2) \alpha n - 5 n - 1$, respectively.

Let $n$ be a positive integer and $\mathcal M$ a set of rational $n \times n$-matrices such that $\mathcal M$ generates a finite multiplicative semigroup. We show that any matrix in the semigroup is a product of matrices in $\mathcal M$…

We prove that for every $n \in \mathbb{N}$ and $\delta>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+\delta}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom,…

We introduce the classes of descendingly flexible and descendingly alternative algebras over an arbitrary field $\mathbb{F}$. We suggest a new method based on the sequence of differences between the dimensions of the linear spans of words,…

Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper…

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds…

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered, if it has a proper prefix which is also a suffix of…

Given an alphabet $S$, we consider the size of the subsets of the full sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that $x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a positive,…

We consider two natural problems about nondeterministic finite automata. First, given such an automaton M of n states, and a length l, does M accept a word of length l? We show that the classic problem of triangle-free graph recognition…

We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we…

Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n)) (Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and lower…

Gerstenhaber's theorem states that the dimension of the unital algebra generated by two commuting $n\times n$ matrices is at most $n$. We study the analog of this question for positive matrices with a positive commutator. We show that the…

We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete…

We show that for any finite-rank free group $\Gamma$, any word-equation in one variable of length $n$ with constants in $\Gamma$ fails to be satisfied by some element of $\Gamma$ of word-length $O(\log (n))$. By a result of the first…