-Tim Cocks is a Reuters correspondent based in Baghdad.-

For the U.S. military, it's the million dollar question -- or rather the $687 billion question, according to a recent estimate of the Iraq war's total cost. Is Iraq now stable enough for them to take a permanent back seat?

The short answer is no one knows. The only way they were ever going to find out was to leave Iraq's own forces to it and hope the whole thing doesn't come tumbling down. They started doing that on Tuesday when they pulled out of Iraqi cities.

It's been an encouraging start. A big bomb in Kirkuk cast a shadow over Iraq's celebrations of its new-found sovereignty, but since then things have been relatively quiet. Militants might try to take advantage by stepping up attacks, but for the moment they seem content with celebrating a "victory" over the occupation -- and setting off the odd bomb, of course.

The United States' coalition partners have for the most part long since departed. British forces handed over southern Iraq to the Americans in April, but since 2007 their 4,000 odd troops left had been largely confined to Basra airport anyway.

And one thing the crystal ball gazers have learned about Iraq's hugely complicated, many-sided conflict is that the past is rarely a reliable guide to the future.

When optimists thought Iraq was poised to enjoy democracy after the fall of Saddam, it spiralled into years of bloody insurgency and sectarian killing. Later, just when it seemed all hope was lost and Iraq would have to be partitioned, things starting getting dramatically better.

The idea that Iraqi forces aren't ready to take on the country's security usually centre on claims that they are untested, not well trained or infiltrated with militiamen.

But few deny they look more professional and integrated now than anyone would have thought possible two years ago. They might still be full of militiamen, but those militiamen are no longer kidnapping or killing political rivals, as in the past.

And there are clearly some things the Iraqis do better. For one thing, they know the language and understand the culture.

When I was on a U.S. patrol in Iraq's troubled Diyala province, a U.S. unit nearby accidentallly shot and wounded a civilian in Jalawla town, forcing them to vacate it because a public outcry would put other soldiers at risk of attack.

What they had done is fire a warning shot at a vehicle after the driver failed to heed a command -- in English -- to stay back. But few Iraqis in rural areas speak basic English.

The real test will be when U.S. pulls all combat forces out, under President Barack Obama's orders, by September next year.

Many Iraqis I've spoken too seem convinced the insurgents are just biding their time, sharpening their knives and stockpiling explosives waiting to reignite the conflict.

But whether or not Iraq can look after itself, at some point the Americans have to say: Look, we've done our best to get the lid back on Pandora's Box. Now it's over to you.

Iraq’s leaders have overcome many hurdles in their struggle to rebuild their country after the ouster of Saddam Hussein in 2003.  But agreeing on the fate of the “ethnic tinderbox” of oil-producing Kirkuk is a particularly testing one.

Why has Kirkuk proven to be such an obstacle? For many, settling its fate seems to be an easy task.

The dispute largely revolves around Kurdish demands to incorporate the city into their autonomous northern Iraq region.  Arabs and Turkmens want the city to remain under the control of the Iraqi government as it has always been.

For an outsider the dispute might seem to be an administrative question of who will manage the city but Kirkuk’s fate has taken on national and regional dimensions since U.S.-led forces toppled Saddam. It has fuelled the ethnic conflict between Arabs
and Kurds and drawn in regional powers, especially neighbouring Turkey.

Kurds look at the city inhabited by Arabs, Kurds and Turkmens as their historic capital, while Arabs and Turkmen argue it equally belongs to them.

While Sunnis and Shi’ite Arabs are locked in a power struggle across the country, they are united in rejecting ceding the city to the Kurdish autonomous region.

But Kirkuk is more than a piece of real estate inheritance. The city sits on a sea of “black gold” — Iraq’s biggest oil field, which has become more lucrative with crude prices above $100 a barrel.

From a regional perspective, Ankara opposes Kurdish control of Kirkuk not only out of concern for the rights of fellow Turkmens in Iraq but also because it will bolster its own Kurdish minority’s demands for autonomy.

Watching an independent Kurdistan gradually taking shape across its border, Ankara fears that Kirkuk’s oil could strengthen the autonomous region in the face of a weak central government in Baghdad, and realise Kurdish aspirations for a region-wide Kurdish state, possibly encompassing southern Turkey and parts of Iran and Syria.

After years of trying and failing, Iraqi leaders are trying to reassure friends and foes that they are close to a deal on the future of Kirkuk. But even if parliament adopts a compromise hammered out behind closed doors, it is difficult to see how it will be implemented.