Ahmes expansions of formal Laurent series and a class of nonarchimedean integral domains

Abstract

Let R be an integral domain with quotient field K . It is proved that each nonzero element fg −1 of K((X)) such that f, g ε R[[X]] and deg( f )<deg( g ) can be expressed as r 1 / h 1 + … + r m h m with each h i , ε R[[X]] ,0≠ r i ε R and {deg( h i )} positive strictly increasing if and only if∩ Rr n ≠ 0 for each nonzero element r of R . If the above conditions hold, then each fg −1 as above admits an expression of the above type with m = 1. The above conditions do hold if K((X)) is the quotient field of R[[X]] . The converse is false even if R is a valuation domain. Indeed, a valuation domain R admits such expansions within K((X)) if and only if each nonzero prime ideal of R has infinite height.