2D function separable into an inner product of 1D vectors

Question

Suppose I have a function $f(x, y) : \mathbb{R}^2 \rightarrow \mathbb{R}$. I want to know when there exist vector functions ${\bf g}(x) : \mathbb{R} \rightarrow \mathbb{R}^d$ and ${\bf h}(y) : \mathbb{R} \rightarrow \mathbb{R}^d$ such that $f(x, y) = {\bf g}(x)^T {\bf h}(y)$. For example, I think a sufficient condition is that $f(x, y)$ is a polynomial in $x$ and $y$. Are there any weaker sufficient conditions? Necessary conditions? If we call $f(x, y)$ $d$-separable when it can be represented by such a form where ${\bf g}(x)$ and ${\bf h}(y)$ map to $\mathbb{R}^d$, can we say anything about the maximum $d$ (e.g., for a polynomial of degree $d$ this is at least $d+1$)?