= Segre Hilbert polynomial and degree
{title2=$P(t)=\binom{t+n}{n}\binom{t+m}{m}$}
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 $t\geq0$. Its <polynomial> degree is $n+m$, giving the <variety>'s <dimension of an algebraic set>[dimension]. Its <leading coefficient> is $1/(n!m!)$, giving embedded projective degree $(n+m)!/(n!m!)=\binom{n+m}{n}$. The two uses of degree therefore describe different quantities.
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