Research on triviality of real singularities

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In this research we have studied triviality of real algebraic singularities and real analytic singularities. In particular. we have considered die problem whether a finiteness theorem holds or not and looked for invariants for some equivalence of singularities. Finiteness property guarantees for the triviality we are considering to be appropriate or reasonable, and the discovery of invariants is important to show that the family of singularities is not trivial.(1) Finiteness theorems on blow-semialgebraic triviality for the family of Nash sets.In the previous paper, we showed that for the family of zero-sets of Nash mappings defined over a compact Nash manifold and the family of zero-set-germs of Nash mappings, we can divide the parameter space into finitely many Nash manifolds on which the family of zero-sets admits a Nash trivial simultaneous resolution. It follows from this result that a finiteness theorem holds on Blow-Nash triviality in the case of isolated singularities. In addit … More ion, we proved also that for the family of Nash surfaces embedded in the 3-dimensional space we can divide tht parameter space into finitely many Nash manifolds on which the family is Blow-semialgebraically trivial in the case of non-isolated singularities.In this research, we have considered the finiteness problems on Blow-semialgebraic triviality for real algebraic singularities in the case of non-isolated singularities. Concerning these problems, we have shown a finiteness theorem for the family of Nash surfaces embedded in the space of general dimension and a finiteness theorem for the family of 3-dimensional Nash sets not necessarily embedded in some space.(2) Introductioin of motivic-type invariants for blow-analytic equivalence.In the joint research with A.Parusinski, we have introduced motivic-type invariants for blow-analytic equivalence through an observation for concrete polynomial functions of 3 variables. This was motivated by Denef-Loeser invariants of complex analytic singularities. We have proved that they are blow-analytic invariants and we have given some formulae to compute our invariants and also the Thom-Sebastiani formulae. In addition, using some blow-analytic triviality theorems and these invariants, we have given a blow-analytic classification of Brieskorn polynomials of 2 variables and a blow-analytic classification of almost all Brieskorn polynomials of 3 variables. Less