Polynomials on vector spaces

A polynomial of degree $\leq p$ from a vector space $V\to \mathbb {R}$ is a map $P:V\to \mathbb {R}$ of the form
$P(v)=\sum _{k=0}^{p}A_{k}(v,\dots ,v)$
for $v\in V$ and $A_{k}\in L_{sym}^{k}(V,\mathbb {R} )=$the symmetric k-linear form for $k=0,\dots ,p$.