On asymptotic values of functions in a polydisk domain and Bagemihl's theorem

Asymptotic sets of functions in a polydisk domain of arbitrary connectivity are studied. We construct an example of such function, having preassigned asymptotic set. This result generalizes well-known examples, obtained by M. Heins and W. Gross for entire functions. Moreover, it is found out that not all results on asymptotic sets of functions in $\mathbb{C}$ can be extended to functions in $\mathbb{C}^n$. In particular, this fact is connected with the failure of Bagemihl's theorem on ambiguous points for functions in $\mathbb{R}^n,$ $n\geq 3$.