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Blow-analytic equisingularities, properties, problems and progress. (English) Zbl 0954.26012
Fukuda, T. (ed.) et al., Real analytic and algebraic singularities. Harlow: Longman. Pitman Res. Notes Math. Ser. 381, 8-29 (1998).
This is a survey article on the subject related to the blow-analytic equivalence for real analytic functions. What is the blow-analytic equivalence? Let \(U\) be an open set in \(\mathbb{R}^n\), and \(g: U\to\mathbb{R}\) be a continuous function. They define that \(g\) is blow-analytic if there exists a composition \(\beta: M\to U\) of a finite number of blowing-ups with smooth centers such that \(g\beta\) is an analytic function. A typical example is \(g= x^2y/(x^2+ y^2)\). Let \(h: (\mathbb{R}^n,0)\to (\mathbb{R}^n, 0)\) be a local homeomorphism. They say that \(h\) is blow-analytic if the components of both \(h\) and \(h^{-1}\) are blow-analytic functions. Given two function germs \(f,f': (\mathbb{R}^n,0)\to (\mathbb{R},0)\) are said to be blow-analytically equivalent, if there is a blow-analytic homeomorphism \(h\) with \(f'= fh\). It is well known that for the classification of real functions, \(C^1\)-equivalence is too fine. For example, in Whitney’s example \(W_t= xy(x- y)(x- ty)\) (\(t\) is a parameter) a family depending on one continuous parameter contains infinitely many \(C^1\)-equivalence classes. To overcome this difficulty T.-C. Kuo proposed the above concept of “blow-analytic equivalence”.
This article would be a very good guide for those who are interested in this subject. In complex analytic category we have Arnold’s classification list of functions with relatively simple structure. In Arnold’s work only the complex analytic coordinate change on the domain of functions is applied to define the equivalence of functions. Is it possible to give the list corresponding to Arnold’s one for real functions and blow-analytic equivalence?
For the entire collection see [Zbl 0882.00014].

26E05 Real-analytic functions
32S15 Equisingularity (topological and analytic)
57R45 Singularities of differentiable mappings in differential topology
58C25 Differentiable maps on manifolds
14P20 Nash functions and manifolds