Segre Hilbert polynomial and degree ($P(t)=\left(\genfrac{}{}{0ex}{}{t+n}{n}\right)\left(\genfrac{}{}{0ex}{}{t+m}{m}\right)$)

For the Segre embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^m$, the homogeneous coordinate ring is the algebra generated by $x_i{y}_j$, assigning degree one to every generator. Its degree-$t$ piece consists of all polynomials bihomogeneous of bidegree $(t,t)$: pair the $t$ factors of an $x$-monomial with the $t$ factors of a $y$-monomial to express any such monomial as a product of generators. Counting the independent monomials gives the displayed Hilbert polynomial, agreeing with the Hilbert function for every integer $t\ge 0$. Its polynomial degree is $n+m$, giving the variety's dimension. Its leading coefficient is $1/(n!m!)$, giving embedded projective degree $(n+m)!/(n!m!)=\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$. The two uses of degree therefore describe different quantities.