Bhargava's factorials in several variables (Q2376689)

\textit{M. Bhargava} [J. Reine Angew. Math. 490, 101--127 (1997; Zbl 0899.13022)] introduced the notion of \(v\)-orderings for an arbitrary subset \(S\) of integral domain \(D\), and used this notion to study polynomial function. Later [Am. Math. Mon. 107, No. 9, 783--799 (2000; Zbl 0987.05003)] he noted the possibility of a generalization to the case of subsets of \(D^n\) for \(n\geq2\). This is done in this paper and is applied to a study of polynomials in \(n\) variables over the field of quotients of \(D\), which map \(S^n\) in \(S\).