In this paper we propose a computational treatment of the resolution of zero pronouns in Japanese discourse, using an adaptation of the centering algorithm. We are able to factor language-specific dependencies into one parameter of the centering algorithm. Previous analyses have stipulated that a zero pronoun and its cospecifier must share a grammatical function property such as {\sc Subject} or {\sc NonSubject}. We show that this property-sharing stipulation is unneeded. In addition we propose the notion of {\sc topic ambiguity} within the centering framework, which predicts some ambiguities that occur in Japanese discourse. This analysis has implications for the design of language-independent discourse modules for Natural Language systems. The centering algorithm has been implemented in an HPSG Natural Language system with both English and Japanese grammars.