When $f(x,y)=x-y$ the map $$(a,b,c)\mapsto (f(a,b),f(b,c),f(a,c))$$ gives a parametrization of $x+y=z$.
More generally, for an arbitrary polynomial $f$ in $k$ variables and an integer $n$ bigger than $k$ we can consider the map \begin{align*}\Lambda(f,n):\mathbb{C}^n&\longrightarrow \mathbb{C}^{n^{\underline{k}}}\\ (x_1,\dots,x_n)&\longmapsto \left\{\left. (f(x_{i_1},\dots,x_{i_k}))\in \mathbb{C}^{n^{\underline{k}}}\right| (i_1,i_2,\dots,i_k)\in [k]^n\setminus \text{diagonals}\right\}. \end{align*} I am interseted in the polynomials vanishing in the image of $\Lambda(f)$ for some $f$. Have someone studied this before?