On the polynomial closure in a valued field

Abstract

Let D be an integral domain with quotient field K. For any subset S of K, the D-polynomial closure of S is the largest subset T of K such that, for every polynomial f in K [ X ] , if f maps S into D then f maps also T into D. When D is not local, the D-polynomial closure is not a topological closure. We prove here that, when D is any rank-one valuation domain, then there exists a topology on K such that the closed subsets for this topology are exactly the D-polynomially closed subsets of K.