Solving equations of length seven over torsion-free groups
Let \(G\) be a group. An expression \[ g_1x^{\varepsilon_1}g_2x^{\varepsilon_2} \dots g_nx^{\varepsilon_n}=1 \quad (g_i\in G, \varepsilon_i=\pm 1) \] \noindent is called equation of length \(n\). This equation is solvable over \(G\) if there is an embedding of \(G\) into a group \(H\) where this equation has a solution. It is shown that all equations of length \(\leq 7\) in torsion-free groups are solvable over these groups. A previously known estimate was \(\leq 6\) [\textit{M. I. Prishchepov}, Int. J. Algebra Comput. 4, No. 4, 575--589 (1994; Zbl 0828.20022)]. This supports \textit{F. Levin}'s [Bull. Am. Math. Soc. 68, 603--604 (1962; Zbl 0107.01803)] conjecture that any equation in a torsion-free group is solvable over this group.
scientific article; zbMATH DE number 6824250
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