Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So $\mathrm{Im}(\varphi)$ is the set of matrices with rank at most $1$.
So I get that $\varphi$ is the morphism of affine spaces corresponding to the morphism $\psi\colon\mathbb{C}[z_{11},\dots, z_{mn}]\to\mathbb{C}[x_1,\dots, x_m,y_1,\dots,y_n]$ sending $z_{ij}\mapsto x_iy_j$. so by the correspondence theorem, the set of polynomials vanishing on $\mathrm{Im}(\varphi)$ is just $\ker\psi$.
Then $\ker\psi$ defines a projective variety, and my question is, what exactly is the Hilbert polynomial of this projective variety?
This is based on something I was reading earlier on this site, so I admit I'm stepping out of my usual area of comfort.