Equations in a free group with coefficients depending on parameters (Q1820240)

\textit{R. C. Lyndon} [Trans. Am. Math. Soc. 96, 445-457 (1960; Zbl 0108.023)] presented an algorithm for the determination of a set of so-called parametric words which are solutions, in a free group, of a given equation with one unknown. Also he proved that the problem of solving every equation with one unknown and coefficients with linear forms in parameters as their exponents is decidable. \textit{G. S. Makanin} [Izv. Akad. Nauk. SSSR, Ser. Mat. 46, No.6, 1199-1273 (1982; Zbl 0511.20019)] proved the algorithmic decidability of any equation in a free group. Continuing a previous line of investigations the author considers equations in a free group with coefficients having linear forms of parameters as their exponents. It is proved that there exists a function which may be shown as an estimate for the parameters. So any such parametric equation is equivalent to a set of equations without parameters and is decidable.