Equations with one unknown in nilpotent groups
In [Algebra Logika 16, 457-471 (1977; Zbl 0411.20021)] the reviewer proved that the solution problem for the coefficient-free equations \(f(x_ 1,x_ 2,...,x_ n)=g\) in the free nilpotent group F of class \(\ell \geq 9\) is unsolvable. The question studied by the author is that of solving equations with one unknown in a finitely generated nilpotent group. He proves: Theorem. There exists a finitely presented nilpotent group G for which the problem of recognizing the solvability of equations with one unknown is undecidable. The proof of this theorem, like that of the above-mentioned result of the reviewer is based on a simulation of Diophantine equations in a nilpotent group. At the IX. All-Union Symp. group theory, Moscow (1984), the author announced the following results: the group G in Theorem may be chosen of class 3 or may be chosen as free nilpotent group of class \(\ell \geq 10^{20}\).
scientific article; zbMATH DE number 3851366