Regular versus singular order of contact on pseudoconvex hypersurfaces

TL;DR

This paper investigates the relationship between singular and regular types of points on pseudoconvex hypersurfaces in complex space, establishing conditions under which they coincide and providing a counterexample in the non-pseudoconvex case.

Contribution

It proves that singular and regular types agree for pseudoconvex hypersurfaces when the regular type is 4, extending previous results for lower types, and presents a non-pseudoconvex example with divergent types.

Findings

Singular and regular types agree for regular type less than 4.

In pseudoconvex hypersurfaces, they also agree at regular type 4.

A non-pseudoconvex example shows the types can diverge, with the singular type infinite.

Abstract

The singular and regular type of a point on a real hypersurface $\mathcal H$ in $\mathbb C^n$ are shown to agree when the regular type is strictly less than 4. If $\mathcal H$ is pseudoconvex, we show they agree when the regular type is 4. A non-pseudoconvex example is given where the regular type is 4 and the singular type is infinite.