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The homogeneous coordinate ring of the Segre image is the subalgebra $k[x_i{y}_j]\subseteq k[x_0,\dots ,x_n,y_0,\dots ,y_m]$, grading each product $x_i{y}_j$ 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 givesIn fact this agrees with the Hilbert function for every integer $t\ge 0$. Its leading coefficient is $1/(n!m!)$, so the two degree quantities areThus the Segre variety has dimension $n+m$ and embedded projective degree $\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$.

This gives the Segre Hilbert polynomial and degree directly from its graded monomials.