Certain real surfaces in $\mathbb{C}^2$ with degenerated CR singularities

In this paper, we study the local polynomial convexity of certain smooth real surfaces in $\mathbb{C}^2$ with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from $\mathbb{C}^2$ to $\mathbb{C}^2$, we obtain a normal form for such surfaces near the origin as $\{(z,w)\in\mathbb{C}^2: w= \overline{z}^k+o(|z|^{k})\}$ or $M_t := \left\{ (z,w)\in\mathbb{C}^2 : w=(z+t\overline{z})^k+o(|z|^k) \right\}$, for some $t>0$, where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\geq 3$. We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\frac{\pi}{k}\right)$. On the other hand, for $0<t<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<t<\min\left\{\sin\left(\frac{\pi}{k}\right),\frac{1}{k-2}\right\}$.