Enriched Riemann Sphere, Morse Stability and Equi-singularity in $\mathcal{O}_2$

TL;DR

This paper extends complex analysis to the enriched Riemann sphere with infinitesimals, generalizes Morse stability, and formulates an equi-singular deformation theorem within the local ring of convergent power series.

Contribution

It introduces the enriched Riemann sphere with infinitesimals, extends Morse stability to this setting, and establishes an equi-singular deformation theorem in $O_2$.

Findings

Enriched Riemann sphere includes infinitesimals via Newton-Puiseux field.

Morse stability theorem is generalized to the enriched setting.

An equi-singular deformation theorem is formulated in $O_2$.

Abstract

The \textit{Enriched Riemann Sphere} $\C P_*^1$ is $\C P^1$ plus a set of \textit{infinitesimals}, having the Newton-Puiseux field $\F$ as coordinates. Complex Analysis is extended to the $\F$-\textit{Analysis} (\textit{Newton-Puiseux Analysis}). The classical \textit{Morse Stability Theorem} is also extended; the \textit{stability idea} is used to formulate an \textit{equi-singular deformation theorem} in $\C\{x,y\}(=\mathcal{O}_2)$.