Four characterizations of real rational double points (Q797644)

[This article was published twice, one times in the book Zbl 0527.00017.] This paper is a sequel to the author's paper ''Fifteen characterizations of rational double points and simple critical points'' [Enseign. Math. 25, 132-163 (1979; Zbl 0418.14020)]. The characterizations of that paper are for complex varieties and complex functions, and involve the Dynkin diagrams \(A_ k\), \(D_ k\) and \(E_ k\). It turns out that the missing Dynkin diagrams \(B_ k\), \(C_ k\), and \(F_ 4\) (but not \(G_ 2)\) correspond to real singularities and real functions, and that a smaller number of similar characterizations are true for these as well. The main theorem of this paper contains four such characterizations: Let \(f:({\mathbb{R}}^ 3,0)\rightsquigarrow(\mathbb{R},0)\) be the germ at the origin of a real analytic function. Then the following are equivalent: (1) The germ f is right-left equivalent to one of the germs given in a certain list. (2) The germ f is simple (in the sense of Arnold). (3) The complexified variety \(f^{-1}(0)\) has a rational singularity at the origin. (4) A resolution of the real variety \(f^{-1}(0)\) is given in a certain list. The proof of the theorem proceeds by direct computation, or by referring to the corresponding theorem in the complex case.