P-orderings: a metric viewpoint and the non-existence of simultaneous orderings

For a prime ideal ℘ and a subset S of a Dedekind ring R, a ℘-ordering of S is a sequence of elements of S with a certain minimizing property. These ℘-orderings were introduced in Bhargava (J. Reine Angew. Math., 490 (1997) 101) to generalize the usual factorial function and many classical results were thereby extended, including results about integer-valued polynomials. We consider ℘-orderings from the viewpoint of the ℘-adic metric on R. We find that the ℘-sequences of S depend only on the closure of S in . When R is a Dedekind domain and R′ is the integral closure of R in a finite extension of the fraction field of R, we relate the ℘-sequences of R and R′. Lastly, we investigate orderings that are simultaneously ℘-orderings for all prime ideals ℘⊂R, and show that such simultaneous orderings do not exist for imaginary quadratic number rings.