Equisingularity in R as Morse Stability in Infinitesimal Calculus

Two seemingly unrelated problems are intimately connected. The first is the equsingularity problem in R: For an analytic family ft : (R , 0) → (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∗? The space R∗ is much smaller than the space used in Non-standard Analysis. Our infinitesimals are analytic arcs, represented by fractional power series, e.g., x = y + · · · , x = y + · · · , x = y + · · · , are infinitesimals at 0 ∈ R, in descending orders. Thus, pt(x) := ft(x, y) := x 4 − txy− y is a family of polynomials over R∗. This family is not Morse stable: a triple critical point in R∗ splits into three when t 6= 0. In our Theorem II, (B) is a trivialization condition which can serve as a definition for equisingular deformation; (A), and (A’) in Addendum1, are criteria, using the stability of “critical points” and the “complete initial form”; (C) is the Morse stability (Remark (1.6)). Theorem I consists of weaker conditions (a), (b), (c). The detailed proofs will appear later. We were inspired by the intriguing discovery of S. Koike ([2]) that the Briançon-Speder family, while blow-analytically trivial, admits no contact order preserving trivialization. The notion of blow-analytic trivialization must be modified; (B) and (b) are options.