Let $k$ be a field, let $n\in\mathbb N$, and set $S:=k[x_0,x_1,\dots,x_n]$ and $A:=k[x_1,\dots,x_n]$. For a nonzero polynomial $f\in A$, let $d=\deg f$ and let $f^h\in S$ denote the homogenization of $f$ to total degree $d$. For $H\in S$, define its dehomogenization on the affine chart $D_+(x_0)$ by
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\begin{align*}
H^{\mathrm{deh}}:=H(1,x_1,\dots,x_n)\in A.
\end{align*}
Conversely, let $F\in S$ be a nonzero [homogeneous polynomial](/page/Homogeneous%20Polynomial). Suppose
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\begin{align*}
F=x_0^rG
\end{align*}
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for some $r\in\mathbb Z_{\ge 0}$ and some nonzero homogeneous polynomial $G\in S$. Put $D:=\deg G$. If the homogenization of $F^{\mathrm{deh}}\in A$ is taken to total degree $D$, then