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On the equivariant blow-Nash classification of simple invariant Nash germs. (English) Zbl 1457.14006

The author develops the theory of real analytic Nash germs invariant under right composition with the action of the group \(G = \mathbb Z/2\mathbb Z\) changing the sign of the first coordinate. The present research focuses on the problem of classification of germs that are simple with respect to the equivariant blow-Nash equivalence (see, [G. Fichou, Ann. Pol. Math. 87, 111–126 (2005; Zbl 1093.14007)]). The author’s approach is mainly based on ideas, methods and tools described in his earlier work (see [F. Priziac, Nagoya Math. J. 222, 100–136 (2016; Zbl 1365.14006)]), where the concept of equivariant blow-Nash equivalence for invariant Nash germs was introduced and equivariant zeta functions, which are invariants for the equivariant blow-Nash equivalence, are constructed. As an application the author computes explicitly the equivariant virtual Poincaré series of the fibers over \(0\), \(-1\) and \(+1\) of the quadratic form \(Q_{p,q}(y)=\sum_{i=1}^p y_i + \sum_{j=1}^q y_{p+j}\) equipped with four different actions of \(G\) (cf. [G. Fichou, Ann. Inst. Fourier 58, No. 1, 1–27 (2008; Zbl 1142.14003)].

MSC:

14B05 Singularities in algebraic geometry
14P20 Nash functions and manifolds
14P25 Topology of real algebraic varieties
32S15 Equisingularity (topological and analytic)
57S17 Finite transformation groups
57S25 Groups acting on specific manifolds
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References:

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