Consider Segre embedding $$ \theta: \mathbb{P}^n \times \mathbb{P}^m \longrightarrow \mathbb{P}^{(m+1)(n+1)-1} $$ $$ (x_0:x_1:\ldots:x_n, y_0:y_1:\ldots :y_m) \mapsto (x_0y_0:\ldots:x_iy_j:\ldots:x_ny_m) $$ Suppose we have a set $X \subset \mathbb{P}^n \times \mathbb{P}^m$ which is a zeroes of polynomial $f(x_0:\ldots:x_n, y_0:\ldots:y_m)$, where $f$ is homogeneous by $x_0:\ldots:x_n$ of degree $r$ and homogeneous by $y_0:\ldots:y_m$ of degree $t$.
The first question is:
Is it true (i am asking just to kill any doubts) that we can write $f$ in the following explicit form:
$$
f = \sum_{I = (i_0, \ldots, i_n), J=(j_0,\ldots , j_m)} a_{IJ} x_0^{i_0}\ldots x_1^{i_n}\ldots y_0^{j_0} \ldots y_m^{j_m}
$$
where the $I, J$ satisfy
$$
i_0 + \ldots + i_n = s
$$
$$
j_0 + \ldots j_m = r
$$
And the second and main question:
How i can write $\theta(X)$ in form $$ F(z_0:\ldots z_{nm+n+m}) = 0 $$ Where $F(z)$ is homogeneous polynomial? I am interested in explicit coordinate look of $F(z)$. Or any another good description of $F(z)$?
Edit: As was mentioned, it's not possible to describe image of set as zero of polynomial. So the question is:
How to describe $\theta(X)$ as zero of family of polynomials?