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In this article, we follow the arguments in a paper of Y-T. Siu to study theeffective termination of Kohn's algorithm for special domains in$\mathbb{C}^{3}$. We make explicit the effective constants and genericconditions that appear there, and we obtain an explicit expression for theregularity of the Dolbeault laplacian for the $\overline{\partial}$-Neumannproblem. Specifically, on a local peudoconvex domain of the special shape \[\Omega:= \bigg\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:\ 2\text{Re}\ z_{3}+\sum_{i=1}^{N}|F_{i}(z_{1},z_{2})|^{2}<0 \bigg\} \] with holomorphic functiongerms $F_{1},\dots,F_{N}\in\mathcal{O}_{\mathbb{C}^{2},0}$ of finiteintersection multiplicity \[ s:=\dim_{\mathbb{C}}\\mathcal{O}_{\mathbb{C}^{2},0} \big/ \langle F_{1},\dots, F_{N} \rangle <\infty, \] we show that an $\varepsilon$-subelliptic regularity for$(0,1)$-forms holds whenever, just in terms of $s$, \[ \varepsilon \geqslant\frac{1}{ 2^{(4s^{2}-1)s+3} s^{2}(4s^{2}-1)^{4} \binom{8s+1}{8s-1}}. \]

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