Geodesic
Morsifications of real plane curve singularities
Journal of Singularities, Tome 18 (2018), pp. 307-328

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A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real morsifications for real plane curve singularities having arbitrary real local branches and pairs of complex conjugate branches satisfying some conditions. This was known before only in the case of all local branches being real (A'Campo, Gusein-Zade). We also discuss a relation between real morsifications and the topology of singularities, extending to arbitrary real morsifications the Balke-Kaenders theorem, which states that the A'Campo-Gusein-Zade diagram associated to a morsification uniquely determines the topological type of a singularity. 
@article{10_5427_jsing_2018_18p,
     author = {Peter Leviant and Eugenii Shustin},
     title = {Morsifications of real plane curve singularities},
     journal = {Journal of Singularities},
     pages = {307--328},
     publisher = {mathdoc},
     volume = {18},
     year = {2018},
     doi = {10.5427/jsing.2018.18p},
     url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2018.18p/}
}
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Peter Leviant; Eugenii Shustin. Morsifications of real plane curve singularities. Journal of Singularities, Tome 18 (2018), pp. 307-328. doi: 10.5427/jsing.2018.18p

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