Explicit calculation of Siu's Effective Termination in Kohn's Algorithm for Special Domains in ${\mathbb{C}}^3$

In this article, we follow the arguments in a paper of Y-T. Siu to study the effective termination of Kohn’s algorithm for special domains in ${\mathbb{C}}^3$. We make explicit the effective constants and generic conditions that appear there, and we obtain an explicit expression for the regularity of the Dolbeault laplacian for the $\bar{\mathrm{\partial }}$-Neumann problem. Specifically, on a local peudoconvex domain of the special shape $$\mathrm{\Omega }:=\{(z_1,z_2,z_3)\in {\mathbb{C}}^3:2\text{Re }{z}_3+\sum _{i=1}^N\mathrm{\mid }{F}_i(z_1,z_2){\mathrm{\mid }}^2<0\}$$ with holomorphic function germs $F_1,\dots ,F_N\in {\mathcal{O}}_{{\mathbb{C}}^2,0}$ of finite intersection multiplicity $$s:={\dim }_{\mathbb{C}}{\mathcal{O}}_{{\mathbb{C}}^2,0}/⟨F_1,\dots ,F_N⟩<\mathrm{\infty },$$ we show that an $\epsilon$-subelliptic regularity for $(0,1)$-forms holds whenever, just in terms of $s$, $$\epsilon ⩾\frac{1}{2^{(4s^2-1)s+3}{s}^2(4s^2-1{)}^4\left(\genfrac{}{}{0px}{}{8s+1}{8s-1}\right)}\mathrm{.}$$