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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References
Blanloeil, V., Oka, M.: Topology of strongly polar weighted homogeneous links. SUT J. Math. 51(1), 119–128 (2015)
Bleher, P., Homma, Y., Ji, L., Roeder, R.: Counting zeros of harmonic rational functions and its application to gravitational lensing. Int. Math. Res. Not. IMRN 8, 2245–2264
Elkadi, M., Galligo, A.: Exploring univariate mixed polynomials of bidegree. In: Proceeding of SNC’2014, pp. 50–58 (2014)
Khavinson, D., Neumann, G.: On the number of zeros of certain rational harmonic functions. Proc. Am. Math. Soc. 134(6), 666–675 (2008)
Khavinson, D., Świa̧tek, G.: On the number of zeros of certain harmonic polynomials. Proc. Am. Math. Soc. 131(2), 409–414 (2003)
Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials. Topology 9, 385–393 (1970)
Oka, M.: Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J. 31(2), 163–182 (2008)
Oka, M.: Non-degenerate mixed functions. Kodai Math. J. 33(1), 1–62 (2010)
Oka, M.: On mixed projective curves. Singularities in Geometry and Topology. IRMA Lectures in Mathematics and Theoretical Physics, vol. 20, pp. 133–147. European Mathematical Society, Zürich (2012)
Oka, M.: Intersection theory on mixed curves. Kodai Math. J. 35(2), 248–267 (2012)
Petters, A.O., Werner, M.C.: Mathematics of gravitational lensing: multiple imaging and magnification. Gen. Rel. Gravit. 42(9), 2011–2046 (2010)
Rhie, S.H.: n-point gravitational lenses with \(5(n-1)\) images (2003). arXiv:astro-ph/0305166
Wilmshurst, A.S.: The valence of harmonic polynomials. Proc. Am. Math. Soc. 126(7), 2077–2081 (1998)
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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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DOI: https://doi.org/10.1007/978-3-319-73639-6_16
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