There really isn't a lot of evidence of pre-columbian contact. Outside the Bering strait region, the only firmly established contacts are that of the Norse. Polynesian contacts may have occurred, but aren't considered definitive. Nothing else is considered remotely plausible. That means there weren't a lot of opportunities for anyone to get American plants back to Europe prior to Columbus.
"there weren't a lot of opportunities for anyone to get American plants back to Europe prior to Columbus"
Thinking about this myself a while ago, I wondered - even if someone went there and back, and even brought something along, how could we tell? That Norse saga had a mythical status at most, and even with that it isn't necessarily clear for a casual listener that it may be about reaching another previously unknown continent. Other groups along the history may have also pulled it off, but then their contemporaries, in order to consemnate the event, had to take their word for it? Even with some seeds as evidence, the world was less connected untill very recently and there were plenty of isolated places, yet relatively close, for strange seeds to be found. After thousand of years, it's hard to tell if an annual plant was brought at some point in time by intercontinental explorers, or migrated by some other non-human related way, or has been local since before continents went appart. Then it was hard to discern the magnitude of a discovery. Even the Norsemen, like the later Spaniard colonists, couldn't tell how big their found body of land was, if the Vinland they reached was an island they haven't yet circumnavigated yet or happened to be a continent spanning south down to well into southern hemisphere. No, for a discovery like this to stick, with clear compiled inventories of imported plants, it had to be done by a government-like entity, that could afford the high risks of loss that an exploration program on the limits of available technology pose, and with enough logistics and cultural development to pursue a wide range of interests (as opposed to small groups, for which keeping track of traffiked plants may be of lesser concern).
The evidence (both linguistic and genetic) currently seems to tilt in favor of the idea that Polynesian contact did occur, as evidenced by the presence of pre-Columbian sweet potato cultivation in Polynesia.
Although natural dispersal of Sweet Potato is a possible explanation, the evidence for natural dispersal is weaker than the evidence for pre-Columbian contact (for instance, the similarity of the Quechua word for sweet potato, "kumar", and the Maori "kumara").
The evidence bar for contacts is very high. My personal position is favorable towards limited (but existing) contacts, but it'd be a lie to say that it's definitively established.
