Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in \mathbb{C}^n: F(\mathbf{x}) = 0 \}$, where $k$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(y_1, \ldots, y_N) = A_1 y_1^d + \ldots + A_N y^d \in k[y_1, \ldots, y_N]$ and some linear forms $L_i(\mathbf{y}) \in k[y_1, \ldots, y_N]$, say $1 \leq i \leq T$, such that $$ V(F) = V(D) \cap V(L_1) \cap \ldots \cap V(L_T). $$ Here $k = \mathbb{Q}, \mathbb{R}$ or $\mathbb{C}$. I seem to recall someone telling me this follows fairly easily from some algebraic geometry (possibly using Veronese embedding) but I couldn't figure it out. Any comments would be appreciated.

Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in k^n: F(\mathbf{x}) = 0 \}$, where $F$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(y_1, \ldots, y_N) = A_1 y_1^d + \ldots + A_N y^d \in k[y_1, \ldots, y_N]$ and some linear forms $L_i(\mathbf{y}) \in k[y_1, \ldots, y_N]$, say $1 \leq i \leq T$, such that $$ V(F) = V(D) \cap V(L_1) \cap \ldots \cap V(L_T). $$ Here $k = \mathbb{Q}, \mathbb{R}$ or $\mathbb{C}$. I seem to recall someone telling me this follows fairly easily from some algebraic geometry (possibly using Veronese embedding) but I couldn't figure it out. Any comments would be appreciated.