Simple Singularities of Functions that are Even or Odd in Each Variable
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
SIMPLE SINGULARITIES OF FUNCTIONS THAT ARE EVEN OR ODD IN EACH VARIABLE N. T. Abdrakhmanova Lomonosov Moscow State University Moscow 119991, Russia [email protected]
E. A. Astashov ∗ Lomonosov Moscow State University Moscow 119991, Russia [email protected]
UDC 512.761.5
We present a classification of simple singularities of analytic functions of many real or complex variables possessing the evenness or oddness property in each variable. Bibliography: 16 titles.
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Introduction
The notion of an equivariant mapping arises while considering manifolds on which the action of a group G is given. Definition 1.1. A mapping f : M → N of G-manifolds is said to be equivariant if for all points x ∈ M and all elements σ ∈ G f (σ · x) = σ · f (x). If the action of the group G on the manifold N is trivial, then the mapping f is referred to as invariant under the action of the group G on the manifold M . On the set of equivariant mappings of two given G-manifolds, one can introduce the equivalence relation. Definition 1.2. Two equivariant mappings f, g : M → N of G-manifolds are said to be equivariant right-equivalent (or R G -equivalent) if there exists an equivariant diffeomorphism Φ : M → M such that g = f ◦ Φ. There are many works studying the R G -equivalence classes of equivariant mappings of real or complex G-manifolds and the normal forms of their representatives. In many cases, certain restrictions are imposed on the “complexity” of the mappings under consideration in terms of ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 11-16. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0827
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codimension, modality, or manifold dimension. Precise definitions of these notions and detailed restrictions can be found in the references listed below. Simple (0-modal) singularities are classified in [1] in the case of the trivial group G. is trivial. Singularities of real-valued functions of codimension at most 8 (among which there are both simple singularities and singularities of modality 1) are classified in [2]. The classification of simple singularities on a manifold with boundary obtained in [3] is equivalent to the classification of simple singularities invariant under the Z2 -action in the first coordinate on the preimage. The so-called corner singularities, i.e., singularities of functions on corners or sets of the form Rp ×Rq0 , with codimension at most 4 are classified in [4]. The classification of simple odd singularities, i.e., singularities that are equivariant simple with respect to nontrivial scalar Z2 -actions on the preimage and image, can be found in [5]. The classification of germs of functions of two and three complex variables that are equivariant simple with respect to all possible nontrivial representations of the group Z3 on Cn (n = 2, 3) and C is contained in [6]–[8]. Necessary conditions for the existence of singular germs of holomorphic functions (Cn , 0) →
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