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

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].