Let $k$ be a field with $\operatorname{char}(k)=0$, let $n \in \mathbb{N}$, let $d \in \mathbb{Z}_{\ge 0}$, and let $f \in k[x_1,\ldots,x_n]$. Then
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\begin{align*}
\sum_{i=1}^n x_i \frac{\partial f}{\partial x_i} = d f
\end{align*}
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in $k[x_1,\ldots,x_n]$, where $\frac{\partial f}{\partial x_i}$ denotes the formal [partial derivative](/page/Partial%20Derivative) with respect to $x_i$, if and only if either $f=0$ or $f$ is a nonzero [homogeneous polynomial](/page/Homogeneous%20Polynomial) of total degree $d$.