Let $V$ be a vector space over a field $\mathbb{K}$ and $\mathcal{O}(V)=\mathrm{Sym}V^*$ its ring of polynomial functions. Given $f_1,\dots,f_n\in V^*$, we have the homogeneous polynomial $F=f_1\cdots f_n\in\mathcal{O}(V)$ which can be seen as a map $V\rightarrow \mathbb{K}$ $$F(v):=f_1(v)\cdots f_n(v).$$ This however will not generalize nicely to odd vector spaces. Alternatively, one can view $F$ as a symmetric multilinear map $V\otimes\cdots\otimes V\rightarrow\mathbb{K}$ given by $$F(u_1,\dots,u_n)=\sum_{\sigma\in S_n}f_1(u_{\sigma(1)})\cdots f_n(u_{\sigma(n)}).$$ This generalizes much better to the odd case. Moreover, in the even case both perspectives have the same information since $F(u_1,\dots,u_n)$ coincides with $D^kF(u_1,\dots,u_n)$ when $F$ is viewed as a function of one variable.
In the odd case, where we have a super vector space $\Pi V$ with ring of polynomial functions $\mathcal{O}(\Pi V)=\mathrm{Alt}V^*$ , the corresponding polynomial $F$ can be interpreted as an antisymmetric multilinear map $V\otimes\cdots\otimes V\rightarrow\mathbb{K}$ given by $$F(v_1,\dots,v_n)=\sum_{\sigma\in S_n}\mathrm{sgn}(\sigma)f_1(v_{\sigma(1)})\cdots f_n(v_{\sigma(n)}).$$ I was wondering, is there a way to interpret this as a function of a single variable much like we did above?
Here are my thoughts so far on the matter: Going back to the even case, let $\{e_1,\dots,e_N\}$ be a basis of $V$ and $f(e_{i_1})\cdots f(e_{i_n})=:F_{i_1\cdots i_n}$. Then, if $v=v^ie_i$ with $v^1,\dots,v^N\in\mathbb{K}$, we have $$F(v)=F_{i_1\cdots i_n}v^{i_1}\cdots v^{i_n}.\tag{1}$$ The trick is to now interpret $v^1,\dots,v^n$ not as the components of an element $v\in V$ but rather as the basis of $V^*$ dual to $\{e_1,\cdots,e_N\}$. Then we have $$F=F_{i_1\cdots i_n}v^{i_1}\cdots v^{i_n},\tag{2}$$ and, in light of (1), some authors even denote this by $F(v)$ and call $F$ a function of $v\in V$. Furthermore, since (2) also makes sense in the odd case, they continue to do this, denoting polynomials on $\Pi V$ as functions $F(v)$ of a variable $v\in\Pi V$. On the one hand, one could simply say this is abuse of notation and that whenever it is said "Let $F(v)=F_{i_1\cdots i_n}v^{i_1}\cdots v^{i_n}$ be a function of $v\in \Pi V$" what is really meant is "let $(v^{1},\dots,v^{N})$ be a basis of $V^*$ and $F=F_{i_1\cdots i_n}v^{i_1}\cdots v^{i_n}$". However, the book From Classical Field Theory to Perturbative Quantum Field Theory goes through a lot of effort to avoid this abuse of notation. Since there are several books on the same topic that do not go through this effort and never seem to mention it explicitely, one would hope there is a deeper interpretation. This must also be related to a question I have always had with respect to whether to use the notation $f$ of $f(x)$ when mentioning an element of $\mathbb{K}[x]$. This must also certainly be related to the idea of functor of points.