Let $k$ be an infinite field, let $V$ be a finite-dimensional $k$-[vector space](/page/Vector%20Space), and let $d \in \mathbb{N}$ such that $d!$ is invertible in $k$. Let $P: V \to k$ be a degree-$d$ [homogeneous polynomial](/page/Homogeneous%20Polynomial) function represented by an element of $\operatorname{Sym}^d(V^\ast)$. Then $P$ determines a unique symmetric $d$-linear form $B: V^d \to k$ satisfying \begin{align*} P(v) = B(v,\ldots,v) \end{align*} for every $v \in V$.