Minimal polynomial decomposition of plane curve branch truncation using a semigroup based algorithm
Year: 2025
Type: article
Abstract: Let Y be an irreducible plane curve germ with branch $$\zeta $$ ζ and s characteristic exponents. We introduce a class of truncation sequences of $$\zeta $$ ζ having finite support. For a given $$({\widetilde{\zeta }}_i)_{i=1,\ldots ,s}$$ ( ζ ~ i ) i = 1 , … , s from this class, we explicitly compute the convex hull of the minimal polynomial $$f_i$$ f i for each germ of plane curve $$Y_i$$ Y i , with branch $${\widetilde{\zeta }}_i$$ ζ ~ i . We investigate the relationships between the semigroup of the $$Y_i$$ Y i ’s, as well as the induced canonical valuations. Additionally, we provide methods for selecting truncation sequences that yield topologically equivalent approximations $$Y_s$$ Y s of Y. The sequence $$({\widetilde{\zeta }}_i)_{i=1,\ldots ,s}$$ ( ζ ~ i ) i = 1 , … , s of $$\zeta $$ ζ provides a unique decomposition of each polynomial $$f_i$$ f i . Given that the minimal polynomial $$f_i$$ f i can be written as a power of $$f_{i-1}$$ f i - 1 plus a tail $$\delta _i$$ δ i , our first decomposition theorem studies properties of the tail. The second decomposition theorem characterizes the decomposition of $$\delta _i$$ δ i and enables its explicit computation. To conclude, a pseudocode algorithm is presented along with an example.
Source: Semigroup Forum
Authors: Joao Cabral, Ana Casimiro
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4408860180