The Puiseux Characteristic of a Goursat Germ

TL;DR

This paper establishes a straightforward formula linking the Puiseux characteristic of Legendrian curve germs to the small growth vector of Goursat germs, revealing a deeper connection between singularity theory and nonholonomic distributions.

Contribution

It provides a simple, explicit formula for the Puiseux characteristic based on the small growth vector of Goursat germs, simplifying previous complex algorithms.

Findings

Derived a simple formula for the Puiseux characteristic.

Connected singularity invariants with nonholonomic distribution theory.

Suggested a deeper theoretical link between singularity theory and distribution geometry.

Abstract

Germs of Goursat distributions can be classified according to a geometric coding called an RVT code. Jean (1996) and Mormul (2004) have shown that this coding carries precisely the same data as the small growth vector. Montgomery and Zhitomirskii (2010) have shown that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves: the Puiseux characteristic. Here we derive a simple formula for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector. The simplicity of our theorem (compared with the more complex algorithms previously known) suggests a deeper connection between singularity theory and the theory of nonholonomic distributions.