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## Archive of Issues

Russia Moscow
Year
2016
Volume
26
Issue
2
Pages
155-159
 Section Mathematics Title On the classification of singularities that are equivariant simple with respect to representations of cyclic groups Author(-s) Astashov E.A.a Affiliations Lomonosov Moscow State Universitya Abstract We consider the problem of classification of function germs $(\mathbb{C}^n, 0)\to(\mathbb{C}, 0)$ that are equivariant simple with respect to various representations of a finite cyclic group $\mathbb{Z}_m$, $m\geqslant 3$, on $\mathbb{C}^n$ and $\mathbb{C}$ up to equivariant automorphisms of $\mathbb{C}^n$. In the case of matching scalar actions of the group it is shown that for $n\geqslant 2$ there exist no equivariant simple function germs. This result is generalized to the cases where the group action in several variables in $\mathbb{C}^n$ coincides with the action of the group on $\mathbb{C}$. In addition, it is shown that in the case of non-matching scalar actions of $\mathbb{Z}_3$ on $\mathbb{C}^2$ and on $\mathbb{C}$ any equivariant simple function germ is equivalent to one of the germs $A_{3k+1}$, $k\in\mathbb{Z}_{\geqslant 0}$. Keywords classification of singularities, simple singularities, group action, equivariant functions UDC 512.761.5 MSC 14B05 DOI 10.20537/vm160201 Received 12 May 2016 Language Russian Citation Astashov E.A. On the classification of singularities that are equivariant simple with respect to representations of cyclic groups, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 155-159. References Arnol'd V.I. Normal forms of functions near degenerate critical points, the Weyl groups of $A_k$, $D_k$, $E_k$ and Lagrangian singularities, Funct. Anal. Appl., 1972, vol. 6, no. 4, pp. 254-272. Arnol'd V.I. Critical points of functions on a manifold with boundary, the simple Lie groups $B_k$, $C_k$, and $F_4$ and singularities of evolutes, Russian Mathematical Surveys, 1978, vol. 33, no. 5, pp. 99-116. Domitrz W., Manoel M., Rios P. de M. The Wigner caustic on shell and singularities of odd functions, Journal of Geometry and Physics, 2013, vol. 71, pp. 58-72. Bruce J.W., Kirk N.P., du Plessis A.A. Complete transversals and the classification of singularities, Nonlinearity, 1997, vol. 10, pp. 253-275. Arnold V.I., Gusein-Zade S.M., Varchenko A.N. Singularities of differentiable maps, Volumes I-II, Boston: Birkhauser, 1985-1988 (Monographs Math., vol. 82-83). Full text