In the 10 days since the death of Leonid I. Brezhnev, the competition between the United States and the emerging Kremlin leadership already shows signs of heightened intensity.

Yuri V. Andropov, Brezhnev's successor, has yet to establish his authority in the depleted Soviet power structure, a process which in the past has brought prolonged internal struggle.

But even while the Soviet hierarchy is being reorganized, the Soviet Union is moving out of the lethargy of the Brezhnev years, determined to display greater resourcefulness in competing with the United States.

China has provided an early windfall for the new Soviet rulers by encouraging speculation during the Brezhnev funeral about an expanding thaw in Sino-Soviet relations.

The Reagan administration has been silent about Peking's turn from strategic alignment with the United States, a turn that has been under way for months. Only when Chinese foreign minister Huang Hua turned up at the funeral to be greeted ostentatiously by Andropov and then have a lengthy talk with Soviet Foreign Minister Andrei Gromyko did U.S. officials begin to concede their chagrin about Peking's policy.

Chinese Premier Zhao Ziyang underscored that direction Friday by saying that "China never attaches itself to any big power." He emphasized at the same time that China's differences with the Soviet Union are far deeper than those with the United States. The expected retirement of Huang, announced the same day, however coincidental, may have further dimmed the luster of the glow surrounding his visit to Moscow.

Nevertheless, on the diplomatic scales, China's overture to the Soviet Union at a time of succession in the Kremlin counts as a significant gain for the Soviet Union, at the expense of the Reagan administration. Whatever the long-term consequences, it nourishes the image that the new Soviet leadership is anxious to project.

Before his death, Brezhnev charged the United States with "dreaming of isolating and weakening economically the U.S.S.R. and its friends," while mounting "an unprecedented arms race."

The Soviet response, he said, would follow two reinforcing tracks: to "broaden interaction with all those who hold dear peace on earth" and "tirelessly strengthen the defenses of our own country . . ."

The goals announced by Brezhnev were decisions of the collective leadership, shared by Andropov and other Politburo members who acquired increasing weight in decision-making during the last five years as Brezhnev's health deteriorated. They now become Andropov's objectives.

What is expected to follow is a more imaginative effort by the new Soviet leadership to exploit divisions between the United States and its allies and other nations leaning toward Washington, to force a change in American policy or to frustrate it.

The interval for obligatory diplomatic courtesies between adversary nations that follows the death of a national leader has been very brief. Both superpowers recognize they are in an accelerated rivalry that will test their propaganda ingenuity as much as their diplomatic skills, and the most volatile issue between them is their nuclear competition.

Behind many of the positioning statements coming from Moscow and Washington is equal awareness of an especially formidable deadline facing both: the end of 1983.

At that point, without an agreement to curb U.S.-Soviet nuclear weapons, the United States and its Western European allies are committed to deploy in Europe an array of Pershing II and cruise missiles that would make the Soviet Union newly vulnerable to sudden attack.

To the Kremlin this represents an ominous, hair-trigger change in the nuclear strategic balance, plus an escalation in the arms race. To the allied governments, it is only countering a Soviet threat from recently installed SS20 missiles targeted on Western Europe.

A double set of U.S.-Soviet negotiations is under way in Geneva to try limiting intercontinental and medium-range nuclear missiles. However, there is no prospect that they will produce an agreement by the end of 1983, without profound changes in the two sides' polarized bargaining demands.

The Andropov leadership, therefore, must decide at a relatively early stage whether to risk an attempt to reach a compromise with the Reagan administration in the arms race. That would require at least some alteration in Soviet defense strategy.

Any significant change in that strategy would appear to assure a profound dispute with Soviet military commanders -- which Soviet premier Nikita Khrushchev once ventured and ultimately had to abandon.

Andropov's obvious alternative would be to gamble that the Soviet Union can raise the nuclear alarm across Europe and the United States to a level that will compel the United States to drop or alter its own deployment plans.

On past performance, that would be the course the Soviets would be most inclined to take, at least until persuaded that the effort is hopeless.

The sagacity and resourcefuless of the former leader of the KGB, as well as his reputed knowledge of western strengths and weaknesses, are likely to be severely plumbed in either event, along with all the capacities of the Reagan administration.

The administration has been at its weakest in maintaining cohesion in the western alliance on central issues in East-West rivalry, ranging from nuclear strategy to trade with the Soviet bloc.

Only after considerable damage to allied interests did the Reagan administration calm the initial storm of public alarm in Europe it unwittingly created in 1981 over its nuclear war-fighting strategy. With its penchant for implying that all opposition to its nuclear policies serves Soviet interests, however, the administration still runs the risk of alienating many Americans as well as Europeans as this issue is reheated.

Secretary of State George P. Shultz, in turn, has succeeded to a degree in bridging the gaping differences between the United States and its allies on trade with the Soviet Union, although this dispute is by no means fully resolved.

Only two weeks ago, before Brezhnev's death, Shultz and other senior administration officials were operating on the premise that Soviet-American relations were stalled in immobility and could be scaled down on the administration's order of urgent priorities. Obviously that is no longer the case.