For a two variable polynomial $f(x,y)$ which is not constant with respect to $y$, we have an algebraic function $g(x)$ such that for some open subset of $\mathbb{C}$, $f(x,g(x)) = 0$. My question is can we write $g(x)$, at least locally, as a composition of polynomials and inverses of polynomials, i.e. $$ g(x) = (p_1^{-1} \circ p_2 \circ p_3^{-1} \circ ... \circ p_n)(x) $$ for polynomials $p_1...p_n$. If so, is there some way to put bounds on $n$ and the degree of the $p$'s if we know $f$?
For my particular application, I only need to know this for invertible algebraic functions $g:\mathbb{R}\rightarrow\mathbb{R}$.