Characterizations of those binary forms which represent unity

Question

I am interested in knowing when a binary form $f(x,y) = f_0 x^d + f_1 x^{d-1} y + \cdots + f_d y^d $ $\in \mathbb{Z}[x,y]$ represents unity, that is there exists $(x_0, y_0) \in \mathbb{Z}^2$ such that $f(x_0, y_0) = 1$. Clearly a sufficient condition is if either $f_0$ or $f_d$ is equal to $1$. For the single variable case, a polynomial $g(x) \in \mathbb{Z}[x]$ represents unity if and only if $h(x) = g(x) - 1$ has an integral root.

Are there any simple characterizations of binary forms which represent unity? It would also be sufficient to say that 'most' binary forms which represent unity do so for trivial reasons, i.e. that either $f_0$ or $f_d$ is equal to $1$.

Answer 1

I don't think simple characterizations exists, since there is an easy construction with all but one coefficients arbitrary.

Set $y=1$ and $x,f_0 \ldots f_{d-1}$ to whatever you like (distinct is allowed). The resulting equation is monic linear in $f_d$, so solve for $f_d$ and you get binary form with all but one $f_i$ arbitrary.

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On second thought, set $x,y$ to whatever you like to get a linear system in $f_i$ and chose any solution of it.