Homogeneous polynomials on strictly convex domains

We consider a circular, bounded, strictly convex domain $\Omega\subset\mathbb{C}^{d}$ with boundary of class $C^{2}$. For any compact subset $K$ of $\partial\Omega$ we construct a sequence of homogeneous polynomials on $\Omega$ which are big at each point of $K$. As an application for any $E\subset\partial\Omega$ circular subset of type $G_{\delta}$ we construct a holomorphic function $f$ which is square integrable on $\Omega\setminus\mathbb{D}E$ and such that $E=E_{\Omega}^{2}(f):=\left\{z\in\partial\Omega: \int_{\mathbb{D}z}\left\vert f\right\vert^{2}d\mathfrak{L}_{\mathbb{D}z}^{2} =\infty\right\}$ where $\mathbb{D}$ denotes unit disc in $\mathbb{C}$.