With just under three months to go until parliamentary elections in Iraq, attention is focusing on the disputed city of Kirkuk.
The area is divided, principally between Arabs and the Kurds, who have their own autonomous region in the north-east of the country. Neither side can agree on who should control it, or indeed how to share the region's vast oil wealth.
In public at least, Kirkuk's police say they don't need the Americans
They gathered before dawn. Outside the local police station, a unit of American soldiers had joined up with an Iraqi police force about 300 strong.
Inside, Capt Gene Palka, the American officer in charge, was giving some last minute advice to his Iraqi counterparts, pouring over a map of the city.
Then, just as the sun began to rise, they set off. Units of Iraqi policemen surrounded an area of Kirkuk, while the Americans, accompanied by more Iraqi forces, began what they call a "clearing up operation", looking for explosives and insurgents.
"They're going into each house," Capt Palka explained. "They're seeing who's here, they check their ID cards, see if they're on their target list. See if there's any illegal weapons or anything like that inside of there. If they don't find anything then they move on."
US forces withdrew from urban areas in most of Iraq in June
It's slow work. More than three hours of searching reveals one sniper rifle that turns out to be a toy. They've also found a pistol and a Kalashnkiov rifle, but both are legal. They haven't found any of the weapons stashes they believe to be in the area.
But they have made some arrests. Two men in particular catch Capt Palka's attention. One, he believes, may be a high-ranking al-Qaeda operative; the other is suspected of belonging to a different insurgent group.
"That's fantastic!" Capt Palka is clearly pleased with the morning's work. "This could be a big disruption of the network here in Kirkuk."
American forces officially withdrew from urban areas in most of Iraq at the end of June, but in Kirkuk they have been quietly continuing their operations.
I would love to find a bunch of high-value guys every day. It takes a lot of patience, a lot of hard work
Capt Gene Palka
Kirkuk is divided between Arabs, Kurds and a smaller number of Turkmens. All three communities claim the city and its surrounding lands as their own. The Americans fear that, as elections loom, insurgents are increasingly exploiting the existing ethnic tensions to destabilise the whole area.
Inside the city itself, Arabs and Kurds and Turkmens have lived alongside one another for centuries. At a bustling street market, Kirkuk gives the impression of a thriving town where life is getting back to normal. Indeed, many locals say they resent the continuing presence of the Americans.
"The Americans said they would withdraw," one man complains, "but they're still here. Their patrols are still in the city. It's not good. They should leave things to the Iraqis."
Publicly, the Iraqi police agree. Maj General Jamal Tahir is the city's chief of police. He says his men are more than capable of taking care of security by themselves.
"If you compare the operations by terrorists in 2007 with 2009, you will see a decrease in the percentage of attacks by terrorists by more than 80%."
But privately, Iraqi forces are still asking the Americans to accompany them on security operations inside the city.
And back at the local police station, there's an illustration of the challenges they face. Biometric tests on the arrested suspects reveal them to be cases of mistaken identity. The operation has produced zero results.
"I would love to find a bunch of high-value guys every day. It takes a lot of patience, a lot of hard work."
Captain Palka is struggling to put a positive spin on it.
"We've been here a long time. This doesn't look like a Hollywood win, but it's still a victory. They've come a long way since 2003."
As darkness falls over the city, the horizon glows red from giant flares coming off the oil-fields. By some estimates, Kirkuk province contains 4% of the world's oil reserves.
The real worry for the Americans is that if they leave, the Arabs and the Kurds could start fighting, splitting the country in two.
So despite security improvements, US forces here are continuing their patrols on a daily basis. They're counting on the fact their very presence will keep the sides apart. But with no end to the dispute in sight, their soldiers could be here for a long time to come.
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