The local structure of the solutions of the multidimensional translation equation (Q1312267)

This paper continues, in a manner of speaking, work by the author, the reviewer and \textit{Z. Moszner} [Result Math. 19, No. 3/4, 195-201 (1991; Zbl 0724.39008)] on the translation equation \(F(F(z,s),t)=F(z,s+t)\), with the initial condition \(F(z,0)=0\), where \(s,t\) are \(m\)-component, \(z\) and the function values \(n\)-component real vectors. Here, however, local solutions are offered on neighbourhoods, with the possible exception of sets nowhere dense in them, and continuous partial differentiability is assumed. Note: In Remark (v) of section 1, the author gives the example \(F(z,t)=z/(1-zt)\) \((m=n=1)\) as not contained in the general solution of the translation equation given by \textit{Z. Moszner} [Aequationes Math. 9, 46-59 (1973; Zbl 0263.94016); ibid. 37, No. 2/3, 267-278 (1989; Zbl 0672.39005)]. This is not defined for \(t=1/z\); however, with \(f(z)=1- (1/z)\), it is elsewhere of the subform \(f^{-1} (f(z)+t)\) analogous to Moszner's solution.