Consider a weighted homogeneous polynomial $f \colon (\mathbb{C}^{n}, \mathbf{0}) \to (\mathbb{C},0)$ with an isolated critical point at the origin and satisfying $\lambda f(z_1, \dots, z_n) = f(\lambda^{\omega_1} z_1, \dots, \lambda^{\omega_n} z_{n})$ for $\lambda \in \mathbb{C}^{\times}$ for some set of rationals $\{\omega_1, \dots, \omega_n \} \subset (0, \frac{1}{2}]$. Let $J_{f} = \langle \partial_{1} f, \dots, \partial_{n} f \rangle$ denote the Jacobi ideal (over $\mathbb{C}$) of $f$ generated by its partial derivatives. Define the Milnor Algebra of $f$ as the quotient $A_{f} = O_{n} / J_{f}$, where $O_{n} = \mathbb{C} \{z_1, \dots, z_n \}$ denotes the polynomial ring of convergent power series about the origin. The conditions on $f$ ensure that $A_{f}$ is finite dimensional with \begin{align} \dim_{\mathbb{C}} A_{f} = \prod_{i = 1}^{n} \left( \frac{1}{\omega_i} - 1 \right). \end{align}
Questions: Given weighted homogeneous polynomials $f$ and $g$, under what necessary and/or sufficient condition(s) is there a weighted homogeneous polynomial $h$ depending only on $f$ and $g$ such that $\mathcal{A}_{h} \cong \mathcal{A}_{f} \oplus \mathcal{A}_{g}$? Is such an $h$ explicitly constructible in terms of $f$ and $g$? If so, how do the weights of $h$ depend on those of $f$ and $g$?
