Suppose we have a map $L$ from $\Bbb{F}^n$ to $\Bbb{F}^m$ ($\Bbb{F}$ is an algebraically closed field) where each coordinate of $L(x_1, \dots, x_n)$ is a homogeneous linear form in $x_1, \dots, x_n$. Then we know that $\operatorname{Im}(L)$ is a linear subspace of $\Bbb{F}^m$, because the image space is closed under scaling by a field constant and closed under addition.
Now suppose instead of homogeneous linear forms, the coordinates of the output were homogeneous degree 2 (or degree $t$) polynomials in the input variables (or more generally, homogeneous degree $t$ polynomials). That is, $L(\vec{x}) = (L_1(\vec{x}), \dots, L_m(\vec{x}))$, where $L_i$ are homogeneous degree 2 polynomials. The image space is still closed under scaling by a field constant. It isn't clear that the sum of two vectors from the image space will still be in image space.
Is there some other such condition that's true now?
Maybe some polynomial $f$ (dependent on the degree of the polynomials $L_i$ that generate the output coordinates, but independent of the specific polynomials) such that "for every $k$ vectors $u_1, \dots, u_k$ in $\operatorname{Im}(L)$, $f(u_1, \dots, u_k)$ is also in $\operatorname{Im}(L)$"? For instance, when we were working with linear forms, $k=2$ and $f(u,v) = u+v$.