Effective vanishing order of the Levi determinant

TL;DR

This paper establishes an effective upper bound on the vanishing order of the Levi determinant at a boundary point of a smooth domain in complex space, based on the domain's D'Angelo q-type and dimension.

Contribution

It provides a new, explicit bound linking geometric boundary properties to the Levi determinant's vanishing order, utilizing Catlin's boundary system and D'Angelo's techniques.

Findings

Provides an explicit upper bound for the Levi determinant's vanishing order.

Connects boundary geometry (D'Angelo q-type) with algebraic properties of the Levi form.

Uses Catlin's boundary system and D'Angelo's methods to derive the bound.

Abstract

On a smooth domain in complex n space of finite D'Angelo q-type at a point, an effective upper bound for the vanishing order of the Levi determinant $\text{coeff}\{\partial r \wedge \dbar r \wedge (\partial \dbar r)^{n-q}\}$ at that point is given in terms of the D'Angelo q-type, the dimension of the space n, and q itself. The argument uses Catlin's notion of a boundary system as well as techniques pioneered by John D'Angelo.