Let $k$ be a field, let $n \in \mathbb{N}$, and let $f \in k[x_1,\ldots,x_n]$. For each $d \in \mathbb{N} \cup \{0\}$, let $k[x_1,\ldots,x_n]_d$ denote the $k$-vector subspace consisting of $0$ and the homogeneous polynomials of degree $d$. Then there exists a unique family $(f_d)_{d \in \mathbb{N} \cup \{0\}}$ with $f_d \in k[x_1,\ldots,x_n]_d$ for every $d$, such that all but finitely many $f_d$ are equal to $0$ and

\begin{align*} f = \sum_{d=0}^{\infty} f_d. \end{align*}

Here the infinite sum is interpreted as the finite sum over the indices $d$ for which $f_d \neq 0$.