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Minimal polynomial decomposition of plane curve branch truncation using a semigroup based algorithm
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Resumo(s)
Let Y be an irreducible plane curve germ with branch ζ and s characteristic exponents. We introduce a class of truncation sequences of ζ having finite support. For a given (ζ~i)i=1,…,s from this class, we explicitly compute the convex hull of the minimal polynomial fi for each germ of plane curve Yi, with branch ζ~i. We investigate the relationships between the semigroup of the Yi’s, as well as the induced canonical valuations. Additionally, we provide methods for selecting truncation sequences that yield topologically equivalent approximations Ys of Y. The sequence (ζ~i)i=1,…,s of ζ provides a unique decomposition of each polynomial fi. Given that the minimal polynomial fi can be written as a power of fi-1 plus a tail δi, our first decomposition theorem studies properties of the tail. The second decomposition theorem characterizes the decomposition of δi and enables its explicit computation. To conclude, a pseudocode algorithm is presented along with an example.
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We thank the referee for the useful and important comments to improve our article. The authors were supported by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications).
Palavras-chave
Singularities Curves Semigroup