Let $f\in k[x\_0,...,x\_n]\_d$ be a degree $d$ homogeneous polynomial in $n+1$ variables. 


Is there a way to associate to $f$ a form $g(y\_1,...,y\_m)$ which is symmetric in the sets of binary variables $y\_i = [y\_0^i:y\_1^i]$ of degree $d\_i$ in each set for a suitable choice of $m$ and $d\_i$?


For instance we could associate to a polynomial $f\in k[x\_0,...,x\_3]\_2$ the form $g(y\_1,y\_2) = \sum\_{0\leq i\_1,i\_2\leq 3}a\_{i\_1,i\_2}(y\_0^1)^{3-i\_1}(y\_1^1)^{i\_1}(y\_0^2)^{3-i\_2}(y\_1^2)^{i\_2}$.



Homogeneous polynomials and symmetric binary forms
Not yet an answer but you can see $f$ as an element of the symmetric power $S^d(V^{\vee})$ for $V$ a vector space of dimension $n+1$. You can take $V=S^n(W)$ with $W$ of dimension 2. That should give something like your representation for $g$. The indices labeling the $x$ basis of $V^{\vee}$ become labels for monomials of degree $n$ in your $y$ variables.


More explicitly, if
$$
f(x\_0,\ldots,x\_n)=\sum\_{i\_1,\ldots i\_d=0}^n f\_{i\_1,\ldots,i\_d}x\_{1\_1}\ldots
x\_{i\_d}
$$
with the (physicsy) tensor $f\_{i\_1,\ldots,i\_d}$ being symmetric, then
the associated form is
$$
g(y^1,\ldots,y^d)=\sum\_{i\_1,\ldots i\_d=0}^n f\_{i\_1,\ldots,i\_d}
(y\_0^1)^{n-i\_1}(y\_1^1)^{i\_1}\ldots
(y\_0^d)^{n-i\_d}(y\_1^d)^{i\_d}\ .
$$
It is of course essential to now see $f$ as a symmetric multilinear form rather than a homogeneous polynomial, i.e., to work with $n$ distinct pairs of binary variables $y$. This is the idea behind the **symbolic method of classical invariant theory** (see my answer to [this MO question](https://mathoverflow.net/questions/259106/explicit-invariant-of-tensors-nonvanishing-on-the-diagonal) for more details).


