  In the distributed function computation problem, dichotomy theorems,
initiated by Han-Kobayashi, seek to classify functions by whether the rate
regions for function computation improve on the Slepian-Wolf regions or not. In
this paper, we develop a general approach to derive converse bounds on the
distributed function computation problem. By using this approach, we recover
the sufficiency part, i.e. the conditions such that the Slepian-Wolf regions
become optimal, of the known dichotomy theorems in the two-terminal distributed
computing. Furthermore, we derive an improved sufficient condition on the
dichotomy theorem in the multiterminal distributed computing for the class of
i.i.d. sources with positivity condition. Finally, we derive the matching
sufficient and necessary condition on the dichotomy theorem in the
multiterminal distributed computing for the class of smooth sources.
