Let M and N be two closed (not necessarily orientable) surfaces, and
f:M → N a continuous map. By definition, the minimal
multiplicity MMR[f] of the map f denotes the minimal integer k
having the following property: f can be deformed into a map g such
that the number ∣g^{1}(c)∣ of preimages of any
point c∈ N under g is ≤ k. We calculate MMR[f] for any map f
of positive absolute degree A(f). The answer is formulated in terms of
A(f), [π_{1}(N):f_{#}(π_{1}(M))], and
the Euler characteristics of M and N. For a map f with A(f)=0, we
prove the inequalities 2≤MMR[f]≤4.
