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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

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1435-5337
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Volume 29, Issue 2

Issues

Decompositions of the higher order polars of plane branches

Evelia R. García Barroso
  • Corresponding author
  • Departamento de Matemáticas, Estadística e I.O., Sección de Matemáticas, Universidad de La Laguna. Apartado de Correos 456, 38200 La Laguna, Tenerife, Spain
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  • De Gruyter OnlineGoogle Scholar
/ Janusz Gwoździewicz
Published Online: 2016-06-15 | DOI: https://doi.org/10.1515/forum-2016-0049

Abstract

In [1] Casas-Alvero found decompositions of higher order polars of an irreducible plane complex analytic curve generalizing the results of Merle. We improve his result obtaining a finer decomposition where we find out a kind of branches that we call threshold semi-roots. The existence of threshold semi-roots is a new phenomenon observed for the higher order polars. The topological type and the number of these branches is determined by the topological type of the original curve.

Keywords: Irreducible plane curve; higher order polar; threshold semi-root

MSC 2010: 32S05; 32S99

References

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    Casas-Alvero E., Higher order polar germs, J. Algebra 240 (2001), 326–337. Google Scholar

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    García Barroso E. R., Gwoździewicz J. and Lenarcik A., Non-degeneracy of the discriminant, Acta Math. Hungar. 147 (2015), no. 1, 220–246. Google Scholar

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    Gwoździewicz J., Ephraim’s pencils, Int. Math. Res. Not. IMRN 2013 (2013), no. 15, 3371–3385. Google Scholar

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    Hefez A., Irreducible plane curve singularities, Real and Complex Singularities (Saõ Carlos 2000), Lecture Notes Pure Appl. Math. 232, Marcel Dekker, New York (2003), 1–120. Google Scholar

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    Koike S. and Parusiński A., Equivalence relations for two variable real analytic function germs, J. Math. Soc. Japan 65 (2013), no. 1, 237–276. Google Scholar

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    Lê D. T., Michel F. and Weber C., Sur le comportement des polaires associées aux germes de courbes planes, Compos. Math. 72 (1989), 87–113. Google Scholar

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    Merle M., Invariants polaires des courbes planes, Invent. Math. 41 (1977), 103–111. Google Scholar

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    Pham F., Déformations equisingulières des idéaux Jacobiens de courbes planes, Proceedings of Liverpool Singularities Symposium II, Lecture Notes in Math. 209, Springer, Berlin (1971), 218–233. Google Scholar

About the article


Received: 2016-02-22

Revised: 2016-04-03

Published Online: 2016-06-15

Published in Print: 2017-03-01


Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2012-36917-C03-01

The first-named author was partially supported by the Spanish Project MTM2012-36917-C03-01 and the second author was partially supported by the Plan Propio de Investigación de la Universidad de La Laguna-2014.


Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 357–367, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0049.

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