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In the standard graded homogeneous coordinate ring $S=k[x_0,\dots ,x_n]$ of ${\mathbb{P}}^n$, the irrelevant ideal isIt vanishes only at the origin of the corresponding affine cone, which does not represent a projective point.

It is the polynomial-ring instance of the irrelevant ideal of a graded ring.

Let $S=k[x_0,\dots ,x_n]$ and let $B=S/I(X)$ be the homogeneous coordinate ring of a nonempty projective variety. Its Hilbert function is $h_X(t)={\dim }_k{B}_t$. For all sufficiently large integers $t$, this function agrees with a polynomial $P_X(t)$, the Hilbert polynomial.

The dimension $r$ is the largest length of a strict chain of nonempty irreducible closed subvarieties of $X$. Equivalently $r=\dim B-1=\mathrm{deg}{P}_X$. The degree of the embedded variety is the integer specified by the leading termGeometrically it is the number of intersection points with a general linear subspace of codimension $r$, counted with intersection multiplicities. The polynomial's degree is the dimension $r$; the variety's degree is the leading coefficient multiplied by $r!$. Keeping those two meanings separate is important in the next computation.

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.

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.