This is the countable noun: one Polynesian, two Polynesians. It works like Canadian or Brazilian, except that Polynesia is not one country but many islands and peoples, so the word covers a very wide group. Because of that width, speakers often name a single island people instead — a Samoan, a Tongan, a Hawaiian — and keep Polynesian for when they mean the wider group or are contrasting it with other Pacific groups such as Micronesians. The capital P never drops.

Used this way, the word is a language name, so it is uncountable and takes no article: she studies Polynesian. It does not name one single language that everybody on the islands speaks; it names a family of languages that resemble one another. Because of that, people who need to be exact usually give the individual language instead — Samoan, Tongan, Maori — and keep Polynesian for the family as a whole or for comparisons between its members.