Let Y be an irreducible plane curve germ with branch \(\zeta \) and s characteristic exponents. We introduce a class of truncation sequences of \(\zeta \) having finite support. For a given \(({\widetilde{\zeta }}_i)_{i=1,\ldots ,s}\) from this class, we explicitly compute the convex hull of the minimal polynomial \(f_i\) for each germ of plane curve \(Y_i\) , with branch \({\widetilde{\zeta }}_i\) . We investigate the relationships between the semigroup of the \(Y_i\) ’s, as well as the induced canonical valuations. Additionally, we provide methods for selecting truncation sequences that yield topologically equivalent approximations \(Y_s\) of Y. The sequence \(({\widetilde{\zeta }}_i)_{i=1,\ldots ,s}\) of \(\zeta \) provides a unique decomposition of each polynomial \(f_i\) . Given that the minimal polynomial \(f_i\) can be written as a power of \(f_{i-1}\) plus a tail \(\delta _i\) , our first decomposition theorem studies properties of the tail. The second decomposition theorem characterizes the decomposition of \(\delta _i\) and enables its explicit computation. To conclude, a pseudocode algorithm is presented along with an example.
