Stability of Newton Boundaries of a Family of Real Analytic Singularities

Let .1; (x ,y) be a real analytic t-parameter family of real analytic functions defined in a neighborhood of the origin in JH.2 . Suppose that J;(x, .1') admits a blow analytic trivilaization along the parameter 1 (see the definition in § I of this paper). Under this condition, we prove that there is a real analytic t-parameter family ar(x, y) with ao(x, .1') = (x, y) and ar(O, 0) = (0, 0) of local coordinates in which the Newton boundaries of .1; (x . y) are stable. This fact claims that the blow analytic equivalence among real analytic singularities is a fruitful relationship since the Newton boundaries of singularities contains a lot of informations on them. In an equisingular problem it is important to determine which equivalence among singularities is the best. It should be as strong as possible if the number of equivalence classes is kept in the admissible range. If we can find an appropriate equivalence relation, we wilrhave a fruitful equisingular problem. In the theory of complex analytic singularities it is well known that the topological equivalence is just so. Oka [7] shows the following: If a complex analytic t-parameter family J;(x , y) (It I « 1) of complex analytic functions in a neighborhood of the origin has a constant Milnor number f.1 at the origin and the Newton boundary of fa intersects the x, y-axes at two points, then there exists a complex analytic tparameter family a,(x, y) of local coordinates with 0',(0, 0) = (0, 0) and ao(x, y) = (x, y) such that the Newton boundaries of J; associated with the local coordinate a/x, y) are stable. Since f.1-constancy and topological constancyof J;-I (0) are equivalent (see [6]), this result also means that the Newton boundaries of a topologically constant family J;(x, y) are stable in some complex analytic family a,(x, y) of local coordinates. It is well known that the Newton boundaries of singularities have a lot of information on them. Hence the result claims that the topological equivalence among complex analytic singularities is strong enough and it is natural that the equisingular problem with respect to this equivalence relation is fruitful. Received by the editors July 15, 1988 and, in revised form, January 18, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 58C27. 57R45. This research is partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 62740065). the Ministry of Education, Science and Culture. 133 © 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use