Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F[X_1,\dots,X_n]$ a polynomial in $k[X_1,\dots,X_n]$.

I am looking for notes, books or surveys detailing the link between the $S$-integers solutions of equations of the type $F[X_1,\dots,X_n]=c$ and quasi-$S$-integral points (following Serre) on the variety $\mathbb P_n(k)\setminus\{F=0\}$.

I am aware that many aspects are trivial (or almost trivial) and I have already worked on the subject by myself. I would like to see if my conclusions are correct, if I am missing something, if there are point of views I ignored, if there are striking examples...

Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$.

I am looking for notes, books or surveys detailing the link between the $S$-integers solutions of equations of the type $F(X_1,\dots,X_n)=c$ and quasi-$S$-integral points (following Serre) on the variety $\mathbb P_n(k)\setminus\{F=0\}$.

I am aware that many aspects are trivial (or almost trivial) and I have already worked on the subject by myself. I would like to see if my conclusions are correct, if I am missing something, if there are point of views I ignored, if there are striking examples...