US Republican presidential front-runner Donald Trump and top challenger Ted Cruz ended a long standing truce in spectacular fashion with bitter exchanges during what may have been Mr Trump's strongest debate performance to date.

The New York billionaire and the Texas senator appeared to have a split decision by the end of the night, a sign that for all the bluster, little took place that could derail Mr Trump from his lead position with a contest in Iowa on 1 February to begin the search for a Republican presidential nominee.

Projections that the debate would be a free-for-all were accurate. Beyond the Trump-Cruz theatrics, US Senator Marco Rubio of Florida turned in a pugnacious performance with attacks on Mr Cruz and New Jersey Governor Chris Christie.

Florida Governor Jeb Bush presented himself as a voice of reason against Mr Trump.

A Google snap poll showed viewers believed  Mr Trump, who in the most recent debates was at times less engaged, won the night with 37.3% to Mr Cruz's 26.6% and Mr Rubio's 12.1%.

Until the Fox Business Channel debate in North Charleston, South Carolina, Mr Trump and Mr Cruz had been friendly because they have both been chasing conservatives of the Tea Party movement and did not want to anger them.

With Mr Trump now needing to fend off Mr Cruz's rise in Iowa, he pushed his charge that Mr Cruz may not be constitutionally qualified to serve as president because he was born in Canada. The US Constitution says only "natural born" citizens can become president of the US.

"Who the hell knows if you can even serve in office?" Mr Trump told Mr Cruz, drawing a scattering of boos in the audience.

Born in Calgary, Alberta, to a US citizen mother and a Cuban father, Mr Cruz accused Mr Trump of bringing up his birthplace simply because Mr Cruz was leading some polls in Iowa.

Mr Cruz said Mr Trump, who led the movement questioning whether the Hawaiian-born President Barack Obama was really from the US, had asked his lawyers to look into the issue of Mr Cruz's birth in September and concluded there were no issues.

"Since September, the Constitution has not changed, but the poll numbers have," Mr Cruz said. "And I recognise that Donald is dismayed that his poll numbers are dropping in Iowa, but the facts and the law here are really clear."

Mr Trump said Democrats would sue if Mr Cruz were on the Republican ticket, putting their party's chances of winning at risk.

Mr Cruz shot back that he had spent many years studying constitutional law: "I'm not going to be taking legal advice from Donald Trump."

Mr Trump called for higher tariffs on Chinese goods if China does not stop engaging in practices to devalue their currency.

He reiterated a call he has made for a 45% tariff to offset devaluation.

He also criticised Japan, saying the country is engaging in similar practices and pointed to competition between tractors made by US company Caterpillar Inc and those made by Japan's Komatsu Ltd.

"They're devaluing their currency, and they're killing our companies," Mr Trump said. "We are letting them get away with it, and we can't let them get away with it."

Mr Bush criticised Mr Trump for calling for higher tariffs, saying that would only prompt retaliation against US-made goods.

"This would be devastating for our economy," Mr Bush said in response. "We need someone with a steady hand being president of the United States."

Mr Rubio said he opposes tariffs on Chinese goods because it would only increase prices for US consumers.

"China doesn't pay the tariff, the buyer pays the tariff. If you send a tie or a shirt made in China into the United States, and an American goes to buy it at the store and there's a tariff on it, it gets passed on in the price to the consumer," Mr Rubio said.

Mr Trump responded that Mr Rubio's approach - to simply build the US economy in response - would take too long.

"You absolutely have to get involved with China, they are taking so much of what we have in terms of jobs, in terms of money," Mr Trump said. "We just can't do it any longer."

Mr Cruz agreed that a tariff would not solve the trade problem with China.

"If we just impose a tariff, they'll put reciprocal tariffs, which will hurt Iowa farmers and South Carolina producers and 20% of the American jobs that depend on exports," Mr Cruz said, referring to two of the early voting states in the presidential nominating contest.