Characterization of norm and quasi-norm forms in S-adic setting

The goal of the present paper is to characterize the norm and quasi-norm forms defined over an arbitrary number field F in terms of their values at the S-integer points, where S is a finite set of valuations of F containing the archimedean ones. In this way we generalize the main result of the recent paper [T5], where the notion of a quasi-norm form is introduced when F = Q and S is a singleton. In complement, we exhibit some relations with problems and results in this area of research.