Bruce, J. W.; Du Plessis, A. A.; Wilson, L. C. Discriminants and liftable vector fields. (English) Zbl 0821.58007 J. Algebr. Geom. 3, No. 4, 725-753 (1994). Let \(f : (M,S) \to (N,0)\) be a complex analytic map-germ, where \(M\) and \(N\) are complex analytic manifolds and \(S\) is finite. Denote by \(\Sigma(f)\) the critical set and by \(\Delta(f) = F(\Sigma(f))\) the discriminant set of \(f\). The authors pose the question of which map-germs \(f\) are determined by their discriminant set up to \(\mathcal A\)- or \(\mathcal R\)- equivalence. In particular, they prove that there are at most finitely many \(\mathcal A\)-classes of \(\mathcal A\)-finite analytic map-germs with target dimension greater than one with given source dimension and given discriminant set.It should be noted that from an earlier work [the second and third author, Math. Z. 190, 163-205 (1985; Zbl 0593.58014)] it follows that there is only one \(\mathcal R\)-class (and, hence, one \(\mathcal A\)-class) of \(\mathcal A\)-finite map-germs with given discriminant set in the cases \(\dim N > 2\) or \(\dim M = 1\), \(\dim N = 2\). Thus, in the paper under review the remaining case \(\dim M \geq \dim N = 2\) has been analyzed completely. In conclusion the methods developed by the authors are applied to the problem of classification of mappings from the plane to the plane. In fact, it turns out that the \(\mathcal A\)-classification is essentially reduced to the \(\mathcal K\)-classification of singular points of plane algebraic curves. Reviewer: A.G.Aleksandrov (Moskva) Cited in 1 Document MSC: 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 57R70 Critical points and critical submanifolds in differential topology 32C07 Real-analytic sets, complex Nash functions 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 32C20 Normal analytic spaces 58A07 Real-analytic and Nash manifolds 58K35 Catastrophe theory Keywords:versal unfoldings; discriminant sets; bifurcation sets; liftable vector fields; quasihomogeneous germs; hidden singularities; \(A\)-, \(R\)- and \(K\)- equivalence; Nash functions; plane-curve singularities Citations:Zbl 0593.58014 PDFBibTeX XMLCite \textit{J. W. Bruce} et al., J. Algebr. Geom. 3, No. 4, 725--753 (1994; Zbl 0821.58007)