Abstract
The standard way to study a compound singularity is to decompose it into the simpler ones using either blow up techniques or appropriate deformations. Among deformations, one distinguishes between miniversal deformations (related to deformations of a basis of the local algebra of singularity) and good deformations (one-parameter deformations with simple singularities coalescing into a multiple one). In concrete settings, explicit construction of a good deformation is an art rather than a science. In this paper, we discuss some cases important from the application viewpoint when explicit good deformations can be constructed and effectively used. Our applications include: (a) an $ n $-dimensional Euler-Jacobi formula with simple and double roots, and (b) a simple approach to the known classification of phase portraits of planar differential systems around linearly non-zero equilibrium.
Highlights
The restrictions on possible values of the determinants are given by the Euler-Jacobi formula which is both elegant and effective for the applications
In a parallel way, according to the Grobman-Hartman Theorem, the local behavior of a differential system near a hyperbolic equilibrium is equivalent to the behavior of the linearized system at the equilibrium
The standard theoretical method allowing one to attack both problems is known under the wordings resolution of singularity/desingularization: under an appropriate transformation/deformation of the given map, one decomposes a compound singularity into the simpler ones. Practical implementation of this method may be based on the usage of blow up techniques, miniversal deformations, good deformations, to mention a few
Summary
The standard theoretical method allowing one (at least, theoretically) to attack both problems is known under the wordings resolution of singularity/desingularization: under an appropriate transformation/deformation of the given map, one decomposes a compound singularity into the simpler ones Practical implementation of this method may be based on the usage of blow up techniques (a change of variables on a deleted neighborhood, see [10, 12, 13, 22, 25]), miniversal deformations (for example, unfoldings related to deformations of a basis of the local algebra of singularity, see [9, 7, 8]), good deformations (one-parameter deformations with simple singularities coalescing into a multiple one, see [18]), to mention a few.
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