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 $\{(z,w)\in {\mathbb{C}}^2:w={\bar{z}}^k+o(\mathrm{\mid }z{\mathrm{\mid }}^k)\}$ or $M_t:=\{(z,w)\in {\mathbb{C}}^2:w=(z+t\bar{z}{)}^k+o(\mathrm{\mid }z{\mathrm{\mid }}^k)\}$, for some $t>0$, where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\ge 3$. 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 $0<t<\min \{\sin \left(\frac{\pi }{k}\right),\frac{1}{k-2}\}$.