In the frigid waters off the west coast of Greenland, six international oil consortiums -- four of them led by American firms -- have been looking for oil since 1975 with no success thus far.

All told, Danish officials say, 13 leases were let for exploration, and 10 of them had been given up by the end of last year. The other three are expected to be terminated in a few months and Danish specialists say the longer range prospects of finding oil appear bleak, at least on the west coast.

But the same may not be true on Greenland's even wilder and more permanently frozen northeastern coast, and the question of Greenland's share of the wealth in any would-be major oil or mineral find is potentially the most volatile issue contained in the home rule measure that has been worked out between Denmark and Greenland.

The reason Greenland's east coast may be more promising is that the geological formations there are similar to those in the North Sea and off the west coast of Norway, where oil was discovered many years ago, according to Peter Bjerrum of the Royal Danish Trading Co. in Greenland.

The Danes, he says, will probably start some very preliminary explorations from the air in that region, possibly next summer. Gunnar Martens, a mineral specialist on Greenland's municipal council, adds, however, that it would be at least five years before any really serious exploration and drilling in that area could be under way unless the Canadians find oil on the Canadian east coast.

Greenland's east coast has a great deal of polar ice so drilling and exploitation would be more difficult and expensive than for North Sea oil.

Nevertheless two major problems are looming here if oil is found.

One is that one of the major Greenland political parties, the left-leaning Siumut Party, opposes the granting of any more concessions, fearing a blowout or oil well accident that could contaminate and spoil Arctic waters in a country where one-third of the population makes a living from fishing.

The new and clearly left-wing Inuit Youth Socialist Party has also announced its opposition, leaving only the remaining major party, the Atassut, in favor of more exploration.

The provisions of the home rule bill on sharing the weath also have made a number of Greenlanders suspicious of Copenhagen.

The bill calls for respect for "national unity and thus also to the interests of the whole nation," meaning the Danish nation, when it comes to mineral resources. Denmark and Greenland will have joint decision-making authority and thus have individual veto power on basic exploration decisions. The splitting up of proceeds should drilling begin, however, is controversial and unsettled.

Initially, proceeds will be used to replace in the Danish treasury the money Denmark now budgets as economic subsidy for Greenland -- which currently amounts to about $250 million a year. Any excess then should be used for the benefit of the entire nation, including Greenland, but the distribution would have to be negotiated, according to the home rule measure.