By Jon Leyne BBC correspondent at the US State Department

Those steely eyes showed as little emotion as ever, but President Putin must be quietly delighted at what he heard from President Bush on the first day of their summit meeting in Washington.

Mr Bush announced the largest, unilateral cuts in nuclear weapons ever contemplated by an American president.

Within the next decade, he said, the American arsenal would shrink from 7,000 warheads to between 1,700 and 2,200 warheads.

America has proposed slashing the number of nuclear warheads

In return? Not much, apparently.

Mr Putin said he would "try to respond in kind".

Yet Russia's rusting missile collection could soon shrink to as few as 1,000 warheads of its own accord.

And on the issue of missile defence, he had nothing to offer.

For weeks Washington has been alive with rumours of a "grand compromise".

American missile cuts in return for Russian agreement to the testing of a missile defence system.

But at the last minute, either the Russians have played hard to get, or the practicalities have defeated them.

Conflicting aims

What they are trying to achieve may simply be impossible.

Washington wants to test a missile defence system.

President Bush wants to establish a missile defence shield

Moscow wants to keep the Anti-Ballistic Missile treaty which explicitly outlaws tests.

Even the most creative lawyers have trouble squaring that circle.

Amending the treaty is out of the question.

With that dilemma unresolved, there are two reasons for Mr Bush's apparent generosity.

His right-wing supporters are opposed, more to the principle of arms control than to the details of any specific deal.

The whole idea that the United States should be tied down by treaties is anathema.

Hence Mr Bush's decision to go ahead with a unilateral missile cut, without waiting for a quid pro quo.

New relationship

But it is also true that the relationship between the two countries has been transformed since 11 September.

President Putin made a very astute call, when he managed to be the first foreign leader to convey his condolences on that terrible day.

President Putin says he will 'try to respond in kind' to America's offer on arms

He has also made real sacrifices - removing his objections to the stationing of US forces in the central Asian republic of Tajikistan, for example.

Washington needs that help.

It also sees longer term benefits in cultivating a new relationship with Russia.

And if, as President Bush keeps declaring, it is time to get rid of the last vestiges of the Cold War, why does the United States still need some 7,000 warheads pointed at Russia?

Stark choice

Nevertheless, the absence of a deal on missile defence would leave Mr Bush with a tricky dilemma.

He would then face the choice between renouncing the Anti-Ballistic Missile treaty - not an attractive option despite American bravado - or compromising on the missile defence scheme.

The two presidents have two days in the congenial surroundings of Mr Bush's ranch to resolve their differences.

The Anti-Ballistic Missile treaty may be only a piece of paper, but at the moment it is looking like one of the most powerful weapons the Russians have got.