= Solution
The <homogeneous coordinate ring> of the Segre <image> is the subalgebra $k[x_iy_j]\subseteq k[x_0,\ldots,x_n,y_0,\ldots,y_m]$, grading each <product> $x_iy_j$ as degree one. Its degree-$t$ part is precisely the space of <polynomials> bihomogeneous of bidegree $(t,t)$.
Indeed, every <product> of $t$ generators has those two degrees. Conversely, any <monomial> of bidegree $(t,t)$ can pair its $t$ individual $x$ factors with its $t$ individual $y$ factors, expressing it as a <product> of generators. The distinct resulting <monomials> are <linearly independent>. Counting them gives
$$
\boxed{P_X(t)=\binom{t+n}{n}\binom{t+m}{m}.}
$$
In fact this agrees with the <Hilbert function> for every integer $t\geq0$. Its <leading coefficient> is $1/(n!m!)$, so the two degree quantities are
$$
\boxed{\deg P_X=n+m,\qquad\deg X=\frac{(n+m)!}{n!m!}=\binom{n+m}{n}.}
$$
Thus the Segre <variety> has <dimension of an algebraic set>[dimension] $n+m$ and embedded projective degree $\binom{n+m}{n}$.
This gives the <Segre Hilbert polynomial and degree> directly from its graded <monomials>.
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