AI 中文总结
研究自由群\(F_r\)中压缩本原性问题，证明对于\(r\geq2\)可在非确定性多项式时间判定，\(r = 2\)时可在确定性多项式时间判定，还表明在固定秩\(r\geq2\)中压缩字共轭类的自同构极小性也可在确定性多项式时间判定。
AI 中文摘要
对于固定整数\(r\geq2\)，我们证明自由群\(F_r = F(x_1,\dots,x_r)\)中的“压缩本原性问题”可在非确定性多项式时间内判定。即对于表示\(F_r\)中元素\(g\)的\(\{x_1,\dots,x_r\}^{\pm1}\)上的直线程序\(\mathcal A\)，判定\(g\)在\(F_r\)中是否为本原元的问题属于\(\mathsf{NP}\)，输入由\(\mathcal A\)的大小衡量。对于\(r = 2\)，此问题可在确定性多项式时间内判定。我们还表明，在每个固定秩\(r\geq2\)中，\(F_r\)中压缩字共轭类的自同构极小性可在确定性多项式时间内判定。
英文摘要
For a fixed integer $r\ge 2$, we prove that the \emph{compressed primitivity problem} in the free group $F_r=F(x_1,\dots,x_r)$ is decidable in non-deterministic polynomial time. That is, for a \emph{straight-line program} $\mathcal A$ over $\{x_1,\dots,x_r\}^{\pm1}$ representing an element $g\in F_r$, the problem of deciding whether $g$ is primitive in $F_r$ belongs to $\mathsf{NP}$, with input measured by the size of $\mathcal A$. For $r=2$, we prove that this problem is decidable in deterministic polynomial time. We also show that, in every fixed rank $r\ge 2$, automorphic minimality of the conjugacy class of a compressed word in $F_r$ is decidable in deterministic polynomial time.
Comments24 pages, no figures
