关于无挠的群上一类任意长度方程的可解性
On solvability of certain equations of arbitrary length over torsion-free groups
摘要：
设$G$是一个非平凡无挠的群，$s(t)=g_{1}t^{ε_{1}}g_{2}t^{ε_{2}} \cdots g_{n}t^{ε_{n}}=1 \; (g_{i} \in G,\ ε_i=\pm 1)$是$G$上不含$t^{-1}g_{i}t^{-1}, \; g_{i} \in G$形式块的方程.在本文中，我们证明了$s(t)=1$有一个解决方案，在$G$提供了一个单一的关系，系数的$s(t)$保持。我们也推广我们的结果，以方程含有较高的权力$t$。后面的方程也与Kaplansky零因子猜想\cite{K}有关。
英文摘要：
Let $G$ be a non-trivial torsion free group and $s(t)=g_{1}t^{ε_{1}}g_{2}t^{ε_{2}} \cdots g_{n}t^{ε_{n}}=1 \; (g_{i} \in G,\ ε_i=\pm 1)$ be an equation over $G$ containing no blocks of the form $t^{-1}g_{i}t^{-1}, \; g_{i} \in G$. In this paper we show that $s(t)=1$ has a solution over $G$ provided a single relation on coefficients of $s(t)$ holds. We also generalize our results to equations containing higher powers of $t$. The later equations are also related to Kaplansky zero-divisor conjecture \cite{K}.
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