Bhargavaʼs factorials in several variables

Bhargava introduced a notion of factorial sequence associated to a subset S of a Dedekind domain D. This sequence generalizes the usual sequence (n!)n⩾0, since it has similar arithmetical properties. He introduced the factorial sequence of a subset S in a local way, thanks to the notion of v-ordering of S. On the other hand, such a sequence may be defined in a global way, thanks to the notion of integer-valued polynomial on S. In this article, we define factorial sequences in several variables using both, integer-valued polynomials with d indeterminates and v-orderings of subsets of Dd. We will see that these factorial sequences still generalize the arithmetical properties of the sequence (n!)n⩾0.