In the holomorphic polygon case, assume $v_{1},\ldots v_{n-1}$ are in the kernel of the extended linearized Cauchy-Riemann operator on $W^{1,2;\mu=\pi}$, such that $\Omega(\cdot,v_{1},\ldots v_{n-1})$ does not vanish identically as a 1-form on $\Sigma$. Then $\deg q-\sum_{1}^{k}\deg p_{k}+k-2\geq 0$. When the equality is achieved, the holomorphic polygon is an immersion up to the boundary with minimal vanishing at the corner, and the zero dimensional moduli space of holomorphic polygons is regular at $u:\Sigma\to X$.