arXiv 2103.146092021-03-26math.COcs.DM

回文长度与幂的缩减

Palindromic Length and Reduction of Powers

Josef Rukavicka

DOI：: 10.1016/j.tcs.2022.07.015

摘要：

给定一个非空的有限字$v$，令$PL(v)$是$v$的回文长度;它意味着连接等于$v$的回文的最小数目。让$v^R$表示$v$的反转。给定一个有限或无限字$y$，令$Fac(y)$表示$y$的所有有限因子的集合，令$maxPL(y)=\max\{PL(t)\mid t\in Fac(y)\}$。设$x$是具有$maxPL(x)=k<\infty$的无限非终极周期字，并且设$u\in Fac(x)$是本原非空因子，使得$u^5$在$x$中是递归的。令$Ψ(x,u)=\{t\in Fac(x)\mid u,u^R\not\in Fac(t)\}\mbox{.}$我们构造一个无限非终极周期字$\overline x$使得$u^5, (u^R)^5\not\in Fac(\overline x)$，$Ψ(x,u)\subseteq Fac(\overline x)$，和$maxPL(\overline x)\leq 3k^3$ .不太正式地说，我们展示了如何减少$u$和$u^R$在$x$的权力，在这样一种方式，回文长度仍然有界。

英文摘要：

Given a nonempty finite word $v$, let $PL(v)$ be the palindromic length of $v$; it means the minimal number of palindromes whose concatenation is equal to $v$. Let $v^R$ denote the reversal of $v$. Given a finite or infinite word $y$, let $Fac(y)$ denote the set of all finite factors of $y$ and let $maxPL(y)=\max\{PL(t)\mid t\in Fac(y)\}$. Let $x$ be an infinite non-ultimately periodic word with $maxPL(x)=k<\infty$ and let $u\in Fac(x)$ be a primitive nonempty factor such that $u^5$ is recurrent in $x$. Let $Ψ(x,u)=\{t\in Fac(x)\mid u,u^R\not\in Fac(t)\}\mbox{.}$ We construct an infinite non-ultimately periodic word $\overline x$ such that $u^5, (u^R)^5\not\in Fac(\overline x)$, $Ψ(x,u)\subseteq Fac(\overline x)$, and $maxPL(\overline x)\leq 3k^3$. Less formally said, we show how to reduce the powers of $u$ and $u^R$ in $x$ in such a way that the palindromic length remains bounded.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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