{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,18]],"date-time":"2025-09-18T10:35:23Z","timestamp":1758191723471,"version":"3.44.0"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T00:00:00Z","timestamp":1742774400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T00:00:00Z","timestamp":1742774400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005855","name":"Universidade Nova de Lisboa","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005855","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Semigroup Forum"],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n Let Y<\/jats:italic> be an irreducible plane curve germ with branch \n \n $$\\zeta $$<\/jats:tex-math>\n \n \u03b6<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and s<\/jats:italic> characteristic exponents. We introduce a class of truncation sequences of \n \n $$\\zeta $$<\/jats:tex-math>\n \n \u03b6<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> having finite support. For a given \n \n $$({\\widetilde{\\zeta }}_i)_{i=1,\\ldots ,s}$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n \n \n \u03b6<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n i<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> from this class, we explicitly compute the convex hull of the minimal polynomial \n \n $$f_i$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for each germ of plane curve \n \n $$Y_i$$<\/jats:tex-math>\n \n \n Y<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, with branch \n \n $${\\widetilde{\\zeta }}_i$$<\/jats:tex-math>\n \n \n \n \u03b6<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We investigate the relationships between the semigroup of the \n \n $$Y_i$$<\/jats:tex-math>\n \n \n Y<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\u2019s, as well as the induced canonical valuations. Additionally, we provide methods for selecting truncation sequences that yield topologically equivalent approximations \n \n $$Y_s$$<\/jats:tex-math>\n \n \n Y<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of Y<\/jats:italic>. The sequence \n \n $$({\\widetilde{\\zeta }}_i)_{i=1,\\ldots ,s}$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n \n \n \u03b6<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n i<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of \n \n $$\\zeta $$<\/jats:tex-math>\n \n \u03b6<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> provides a unique decomposition of each polynomial \n \n $$f_i$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Given that the minimal polynomial \n \n $$f_i$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can be written as a power of \n \n $$f_{i-1}$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n \n i<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> plus a tail \n \n $$\\delta _i$$<\/jats:tex-math>\n \n \n \u03b4<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, our first decomposition theorem studies properties of the tail. The second decomposition theorem characterizes the decomposition of \n \n $$\\delta _i$$<\/jats:tex-math>\n \n \n \u03b4<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and enables its explicit computation. To conclude, a pseudocode algorithm is presented along with an example.<\/jats:p>","DOI":"10.1007\/s00233-025-10516-3","type":"journal-article","created":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T22:26:30Z","timestamp":1743027990000},"page":"362-391","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Minimal polynomial decomposition of plane curve branch truncation using a semigroup based algorithm"],"prefix":"10.1007","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8857-6909","authenticated-orcid":false,"given":"Joao","family":"Cabral","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6001-6803","authenticated-orcid":false,"given":"Ana","family":"Casimiro","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,24]]},"reference":[{"issue":"3","key":"10516_CR1","doi-asserted-by":"publisher","first-page":"575","DOI":"10.2307\/2372643","volume":"77","author":"S Abhyankar","year":"1955","unstructured":"Abhyankar, S.: On the ramification of algebraic functions. 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