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Northeastern Brazilian Meeting on Singularities

Brazil-Mexico Meeting on Singularities

NBMS 2015, BMMS 2015: Singularities and Foliations. Geometry, Topology and Applications pp 489–511Cite as

On the Roots of an Extended Lens Equation and an Application

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 222))

Abstract

We consider zero points of a generalized Lens equation \(L(z,{\bar{z}})={\bar{z}}^m-{p(z)}/{q(z)} \) and also harmonically splitting Lens type equation \(L^{hs}(z,{\bar{z}})=r({\bar{z}})-p(z)/q(z)\) with \(\deg \, q(z)=n,\,\deg \,p(z)\le n\) whose numerator is a mixed polynomials, say \(f(z,{\bar{z}})\), of degree \((n+m; n,m)\). To such a polynomial, we associate a strongly mixed weighted homogeneous polynomial \(F(\mathbf{z},{\bar{\mathbf{z}}})\) of two variables and we show the topology of Milnor fibration of F is described by the number of roots of \(f(z,{\bar{z}})=0\).

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Authors and Affiliations

  1. Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Mutsuo Oka

Authors
  1. Mutsuo Oka
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Correspondence to Mutsuo Oka .

Editor information

Editors and Affiliations

  1. ICMC, University of São Paulo, São Carlos, São Paulo, Brazil

    Raimundo Nonato Araújo dos Santos

  2. Department of Mathematics, Federal University of Paraíba, João Pessoa, Paraíba, Brazil

    Aurélio Menegon Neto

  3. Mathematics Institute, University of Warwick, Coventry, United Kingdom

    David Mond

  4. ICMC, University of São Paulo, São Carlos, São Paulo, Brazil

    Marcelo J. Saia

  5. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, Mexico

    Jawad Snoussi

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Oka, M. (2018). On the Roots of an Extended Lens Equation and an Application. In: Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015 2015. Springer Proceedings in Mathematics & Statistics, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-319-73639-6_16

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