  The seminal concept of characteristic polygon of an embedded algebroid
surface, first developed by Hironaka, seems well suited for combinatorially
(perhaps even effectively) tracking of a resolution process. However, the way
this object evolves through the resolution of singularities was not really well
understood, as some references had pointed out. The aim of this paper is to
explain, in a clear way, how this object changes as the surface gets resolved.
In order to get a precise description of the phenomena involved, we need to use
different techniques and ideas. Eventually, some effective results regarding
the number of blow-ups needed to decrease the multiplicity are obtained as a
side product.
