New Yorkers go to the polls with one question: can Rudy run again?

At the High School for Environmental Studies on 56th Street and 10th Avenue they were arguing with smiles on their faces in the drizzle of a grey New York autumn yesterday. It was a good argument, timely and distracting and focused on something very important to Americans at the moment – democracy.

This was the re-run of the mayoral primaries, elections suspended on 11 September in the dark hours after the World Trade Centre terrorist attack, but the election landscape was now completely different. When voters went to the polls two weeks ago, the question was which of the four Democrats and two Republicans would be chosen to represent their parties for the formal battle on 6 November.

Now all of New York was asking: Can Rudy go for a third term? Under election laws introduced in 1993, the mayor must stand down after two four-year terms. Mayor Giuliani, who is being treated for prostate cancer, has completed his time in office but his rock-steady performance during the terrorist crisis has endeared him to New Yorkers, giving him a 95 per cent approval rating and raising in many the hope that a way can be found for him to stand again.

The argument is complex, fraught with legal difficulties and confusing for voters. A poll yesterday found that 75 per cent would like Mayor Giuliani to stand for election again, while 52 per cent said term limits should not be abolished. It was a paradox that exercised the minds of voters at the high school on 56th Street, a polling station in mid-town Manhattan.

"Absolutely, he should be mayor again," said Nikbin Houshang, 60. "He's cleaned up the city, put the city economy back on its feet and brought crime down. And his performance during the past two weeks has been magnificent. They should do whatever they can to get the law changed and get him back in."

It is true that during Mr Giuliani's term, crime has fallen by 60 per cent, unemployment has been slashed and the number of people claiming welfare has halved. It is also true, however, that only a few months ago, the Mayor's very public split from his wife, Donna Hanover, over his living arrangements while conducting an affair with his girlfriend, Judith Nathan, had made him a laughing stock.

Charles Kochman, 39, an editor, said: "There is no way they should allow him to stand. If we agree to let him go for a third term, then we set a very dangerous precedent and the next thing you know, we'll have a president going for a third term."

Many voters were exercising their right to "write in" the name of a candidate not on the ballot papers. A confusing affair that involves pressing a button on the city's complex voting machines, opening a slot and scribbling down, in this case, Rudy Giuliani's name. They can do the same on 6 November, even if Mr Giuliani is not standing officially. That could create a legal nightmare that would make last year's presidential election conundrum look simple.

Valerie Marsch, 38, a television presenter, said: "Writing in Giuliani's name is something I wouldn't do. This is our system and we should stick to it. But if they changed that system, if they altered the law to say Rudy Giuliani could stand and we should all come back tomorrow and vote again, then I'd be down here like a shot to vote for him. And I think most of New York would, too."

There are several ways Mr Giuliani could circumvent the legislation. The state legislature could change the law or a city council bill could be introduced to repeal the law, but both these options are unlikely.

Mr Giuliani, a Republican, could be put forward to replace a current Conservative candidate, Michael Long, a lawyer. Mr Long could stand down only if he were convicted of a felony, moved out of the state or if he put himself forward for appointment as a Supreme Court judge, something he said he was prepared to consider. But all of these would still require a change in the term-limit laws.