# Jet vanishing orders and effectivity of Kohn's algorithm in dimension $3$

## TL;DR

This paper introduces jet vanishing orders as geometric invariants to improve Kohn's algorithm for subelliptic estimates in dimension 3, providing explicit bounds, stability under perturbations, and demonstrating sharpness through examples.

## Contribution

It develops a new selection algorithm using jet vanishing orders, achieving effective termination with explicit bounds and enhanced stability, advancing the understanding of subelliptic multipliers in complex analysis.

## Key findings

- Effective bounds for algorithm steps and subellipticity order
- Stability under high order perturbations demonstrated
- Complete procedure illustrated for perturbations of known examples

## Abstract

We propose a new class of geometric invariants called jet vanishing orders, and use them to establish a new selection algorithm in the Kohn's construction of subelliptic multipliers for special domains in dimension $3$, inspired by the work of Y.-T. Siu [S10]. In particular, we obtain effective termination of our selection algorithm with explicit bounds both for the steps of the algorithm and the order of subellipticity in the corresponding subelliptic estimates. Our procedure possesses additional features of certain stability under high order perturbations, due to deferring the step of taking radicals to the very end of the algorithm. We further illustrate by examples the sharpness in our technical results and demonstrate the complete procedure for arbitrary high order perturbations of the Catlin-D'Angelo example [CD10] in Section 5. Our techniques here may be of broader interest for more general PDE systems, in the light of the recent program initiated by the breakthrough paper of Y.-T. Siu [S17].