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Home > Vol 30, No 2 (2009)

Keith Johnson

If R is a Dedekind domain, P a prime ideal of R and S\subseteq  R a finite subset then a P-ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101-127,  1997), is an ordering { a _i } _i=1 ^m of the elements of S with the property that, for each 1< i\leq  m, the choice of a _i minimizes the P-adic valuation of \prod  _{j< i} ( s -  a _j ) over elements s\in  S. If S, S ^{$^{\prime}$} are two finite subsets of R of the same cardinality then a bijection φ : S\rightarrow  S ^{$^{\prime}$} is a P-ordering equivalence if it preserves P-orderings. In this paper we give upper and lower bounds for the number of distinct P-orderings a finite set can have in terms of its cardinality and give an upper bound on the number of P-ordering equivalence classes of a given cardinality.