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

For a prime ideal Weierstrass p and a subset S of a Dedekind ring R, a Weierstrass p-ordering of S is a sequence of elements of S with a certain minimizing property. These Weierstrass p-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 Weierstrass p-orderings from the viewpoint of the Weierstrass p-adic metric on R. We find that the Weierstrass p-sequences of S depend only on the closure of S in image. 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 Weierstrass p-sequences of R and R′. Lastly, we investigate orderings that are simultaneously Weierstrass p-orderings for all prime ideals Weierstrass psubset ofR, and show that such simultaneous orderings do not exist for imaginary quadratic number rings.