$P$-orderings of finite subsets of Dedekind domains

Johnson, Keith

Journal of Algebraic Combinatorics, Tome 30 (2009) no. 2, pp. 233-253.

Résumé

Summary: 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 varphi: SrightarrowS ^^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.

Détail

Keywords: keywords $P$-ordering, $P$-sequence, Dedekind domain