SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1252-1267, January 2021.
Let $n$ be a natural number, and let $\mathcal{M}$ be a set of $n \times n$-matrices over the nonnegative integers such that the joint spectral radius of $\mathcal{M}$ is at most one. We show that if the zero matrix $0$ is a product of matrices in $\mathcal{M}$, then there are $M_1, \ldots, M_{n^5} \in \mathcal{M}$ with $M_1 \cdots M_{n^5} = 0$. This result has applications in automata theory and the theory of codes. Specifically, if $X \subset \Sigma^*$ is a finite incomplete code, then there exists a word $w \in \Sigma^*$ of length polynomial in $\sum_{x \in X} |x|$ such that $w$ is not a factor of any word in $X^*$. This proves a weak version of Restivo's conjecture.

中文翻译：

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关于非负整数矩阵和短杀戮词

SIAM Journal on Discrete Mathematics，第 35 卷，第 2 期，第 1252-1267 页，2021 年 1 月。
令 $n$ 为自然数，并令 $\mathcal{M}$ 为一组 $n \times n$-矩阵在非负整数上，使得 $\mathcal{M}$ 的联合谱半径至多为 1。我们证明如果零矩阵 $0$ 是 $\mathcal{M}$ 中矩阵的乘积，那么有 $M_1, \ldots, M_{n^5} \in \mathcal{M}$ 和 $M_1 \ cdots M_{n^5} = 0$。这一结果在自动机理论和代码理论中都有应用。具体来说，如果$X \subset \Sigma^*$是一个有限不完全码，那么在$\sum_{x \in X} |x|$中存在一个长度多项式的词$w \in \Sigma^*$ 这样$w$ 不是 $X^*$ 中任何单词的因数。这证明了 Restivo 猜想的弱版本。