Let G be a nontrivial torsion-free group and $s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)$ 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 $s\left( t \right) = 1$ 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.

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关于任意长度的某些方程在无扭转群上的可解性

让G是一个非平凡的无扭群，并且$s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)$是一个方程G不包含表单的块${t^{- 1}}{g_i}{t^{ - 1}},{g_i} \in G$. 在本文中，我们表明$s\left(t\right) = 1$有一个解决方案G提供了关于系数的单一关系s(吨) 成立。我们还将我们的结果推广到包含更高幂的方程吨. 后面的方程也与卡普兰斯基零除数猜想有关。