Abstract We prove the algorithmic undecidability of the problem: whether there exists a solution of an equation w ( x 1 , … , x n ) = [ a , b ] in a free group F 2 of rank 2 with free generators, a and b, with constraints of the type x 1 ∈ F 2 ( 2 ) , where w ( x 1 , … , x n ) is a word in the alphabet of unknowns, { x 1 , … , x n } ; [ a , b ] is the commutator of the free generators, a and b; F 2 ( 2 ) is its second derived subgroup. We also build a polynomial algorithm to recognize whether an arbitrary explicit (i.e. resolved with respect to the unknowns) equation of the type w ( x 1 , … , x n ) = g ( a , b ) , has a solution satisfying the constraint x 1 ∈ F 2 ( s ) , … , x t ∈ F 2 ( s ) , where g ( a , b ) is an element of length less than 4; t is an arbitrary fixed number between 1 and n, and F 2 ( s ) is s-th derived subgroup of F 2 for arbitrary s. The algorithmic decidability of analogous problems for equations with one unknown is also established.

中文翻译：

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自由群中方程兼容性问题的算法不可判定性：具有一个换向器类型约束的显式方程

摘要 我们证明了该问题的算法不可判定性：在具有自由生成器 a 和 b 的秩为 2 的自由群 F 2 中，是否存在方程 w ( x 1 , … , xn ) = [ a , b ] 的解，具有 x 1 ∈ F 2 ( 2 ) 类型的约束，其中 w ( x 1 , … , xn ) 是未知数字母表中的一个词， { x 1 , … , xn } ；[ a , b ] 是自由发生器 a 和 b 的换向器；F 2 (2) 是它的第二个派生子群。我们还构建了一个多项式算法来识别类型为 w ( x 1 , … , xn ) = g ( a , b ) 的任意显式（即相对于未知数求解）方程是否具有满足约束 x 1 的解∈ F 2 ( s ) , … , xt ∈ F 2 ( s ) , 其中 g ( a , b ) 是长度小于 4 的元素；t 是介于 1 和 n 之间的任意固定数，并且 F 2 ( s ) 是 F 2 的第 s 个派生子群，用于任意 s。还建立了具有一个未知数的方程的类似问题的算法可判定性。