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For a coherent sheaf $\mathcal{F}$ of support dimension $d$ on a projective scheme and a fixed Cartier divisor $D$, $h^0(X,\mathcal{F}(mD))=O(m^d)$. Choose a sufficiently high ample divisor $A$ whose section avoids the associated points of $\mathcal{F}$ and for which $D+A$ is ample. Multiplication by the $m$th power of that section injects $\mathcal{F}(mD)$ into $\mathcal{F}(m(D+A))$. The ample Hilbert polynomial, together with Serre vanishing, bounds the latter section space by $O(m^d)$.

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.