A time-invariant fluid model of a polling system is considered. It consists of finitely many servers and buffers with unlimited sizes. The buffers receive inflows of work from the outside, work leaves the system after processing by a server. Every server works only with buffers from an associated zone of service, which may overlap for various servers, is able to serve at most one buffer at a time and so has to switch, from time to time, among buffers, the switch-over times are nonzero. We present a criterion for existence of a scheduling and service protocol that makes the system stable in the sense that the total amount of work in the buffers remains bounded as time progresses. The necessity part of this result is concerned with the widest class of protocols, including dynamic ones that are centralized and have access to the full information about the events in the system. Meanwhile, we show that every stabilizable system can be stabilized in a fully decentralized fashion via a simple static protocol, e.g., by a protocol that is based on independent round robin scheduling of the servers and for every server, employs only time measurement.