Stability of Newton boundaries of a family of real analytic singularities

Abstract:

Let ${f_t}(x,y)$ be a real analytic $t$-parameter family of real analytic functions defined in a neighborhood of the origin in ${\mathbb {R}^2}$. Suppose that ${f_t}(x,y)$ admits a blow analytic trivilaization along the parameter $t$ (see the definition in $\S 1$ of this paper). Under this condition, we prove that there is a real analytic $t$-parameter family ${\sigma _t}(x,y)$ with ${\sigma _0}(x,y)=(x,y)$ and ${\sigma _t}(0,0)=(0,0)$ of local coordinates in which the Newton boundaries of ${f_t}(x,y)$ 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.