Here is a quote from the document linked below:

Abstract Examination of an extensive literature has revealed conclusive evidence that nearly one hundred species of plants, a majority of them cultivars, were present in both the Eastern and Western Hemispheres prior to Columbus' first voyage to the Americas. The evidence comes from archaeology, historical and linguistic sources, ancient art, and conventional botanical studies. Additionally, 21 species of micro-predators and six other species of fauna were shared by the Old and New Worlds. The evidence further suggests the desirability of additional study of up to 70 other organisms as probably or possibly bi-hemispheric in pre-Columbian times. This distribution could not have been due merely to natural transfer mechanisms, nor can it be explained by early human migrations to the New World via the Bering Strait route. Well over half the plant transfers consisted of flora of American origin that spread to Eurasia or Oceania, some at surprisingly early dates. The only plausible explanation for these findings is that a considerable number of transoceanic voyages in both directions across both major oceans were completed between the 7th millennium BC and the European age of discovery. Our growing knowledge of early maritime technology and its accomplishments gives us confidence that vessels and nautical skills capable of these long-distance travels were developed by the times indicated. These voyages put a new complexion on the extensive Old World/New World cultural parallels that have long been controversial.

We do not actually know that Polynesians ever made it all the way to South America.  So far as I know, the strongest evidence so far of this is that the bones of some chickens that seem to be Polynesian have been found in South America.

Hawaii is much easier.  Anyone who knows anything about the Hawaiian language can tell that it is a Polynesian language.  If you go to Hawaii and find that you have natives who look like Polynesians and speak a Polynesian language and have legends about coming from the Southwest, it is clear that those natives are Polynesians.