Invariant polynomials

Let \(V\) be a vector space over \(\mathbb{R}\). Let \(\{e_1,\cdots,e_r\}\) be a basis of \(V\) over \(\mathbb{R}\) and \(\{e^1,\cdots,e^r\}\) be the dual basis of \(V\). We call \(e^i:V\rightarrow \mathbb{R}\) to be polynomials over \(V\) with values in \(\mathbb{R}\).  A map \(p:V\rightarrow \mathbb{R}\) is said to be a polynomial map if

\(p=\sum a_{t_1,\cdots,t_r}(e^1)^{t_1}\cdots(e^r)^{t_r}\).

Let \(f:V\times V\times \cdots\times V\rightarrow \mathbb{R}\) be a symmetric multilinear mapping. We want to associate