Cartan equivalences for Levi-nondegenerate hypersurfaces M^3 in C^2 belonging to General Class I

TL;DR

This paper provides a detailed computational approach to the Cartan equivalence problem for Levi-nondegenerate hypersurfaces in C^2, explicitly deriving the unique essential invariant J and its expansions for general and rigid cases.

Contribution

It offers a comprehensive computational framework for the Cartan equivalence problem, explicitly calculating the invariant J for hypersurfaces in C^2, including the rigid case with simplified expressions.

Findings

Explicit formula for the invariant J in terms of the defining function ta.

The invariant J involves millions of differential monomials in the general case.

Simplified form of J in the rigid case with only 7 monomials.

Abstract

We develope in great computational details the classical Cartan equivalence problem for Levi-nondegenerate C^6-smooth real hypersurfaces M^3 in C^2, performing all calculations effectively in terms of a (local) graphing function \varphi. In particular, we present explicitly the unique (complex) essential invariant J of the problem. Its expansion in terms of the 3-variables function \varphi incorporates millions of differential monomials, while, when \varphi is assumed to depend only on 2 variables (rigid case), J writes out in two lines (7 monomials).