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 [25], where the notion of a quasi-norm form is introduced when \(F = {\mathbb {Q}}\) and S is a singleton. In complement, we exhibit some relations with problems and results in this area of research.