Equisingularity in $R^2$ As Morse Stability

Abstract

Two seemingly unrelated problems are intimately connected. The first is the equsingularity problem in $\R^2$: For an analytic family $f_t:(\R^2,0)\rar (\R,0)$, when should it be called an ``equisingular deformation"? This amounts to finding a suitable trivialization condition (as strong as possible) and, of course, a criterion. The second is on the Morse stability. We define $\R_*$, which is $\R$ "enriched" with a class of infinitesimals. How to generalize the Morse Stability Theorem to polynomials over $\R_*$?