DETROIT, JAN. 15 -- Vice President Bush emerged today with at least a decisive public relations victory -- and perhaps a substantial political coup -- in his battle for this state's 77 national GOP convention delegates, the first to be picked in the nation.

"It's a clear victory," Bush declared, as the coalition of Rep. Jack Kemp (R-N.Y.) and Pat Robertson that has been opposing his campaign in Michigan began to show signs of intensified internal bickering that could end their alliance.

The morning brought a new host of bitter disputes over the results from 124 county conventions held across the state last night. Those conventions chose 1,805 state convention delegates who, in turn, will determine on Jan. 29-30 the makeup of the Michigan delegation to the GOP National Convention in New Orleans.

While Bush's apparent victory may be overturned in a credentials process set up by the Robertson-Kemp controlled state central committee, the results effectively guarantee that Robertson can no longer point to Michigan as an example of how his forces could beat a sitting vice president in a state Bush carried over Reagan by a 2-to-1 margin in 1980.

In addition, the outcome last night was a demonstration of the dogged intensity of the Bush machine which is loaded with a cast of experienced professional strategists and more than enough cash to keep pushing every angle, maneuver and deal when defeat appears inevitable.

Throughout 1987, the Bush drive appeared headed toward sure failure here as the Robertson-Kemp coalition took over the state GOP and repeatedly rubbed Bush's nose in the dirt in rules decisions and procedural fights.

During all these battles, however, the Bush campaign continued to offer secret deals to key Kemp supporters, to set up a sophisticated computerized set of plans that gerrymandered local districts to shift the balance of power and catch Robertson off-guard and to offer all the perquisites of the vice presidency from exclusive limousine trips to fund-raising appearances to persuade aspiring politicians to switch their loyalties to Bush.

Robertson yesterday announced: "I am claiming victory," but he denounced "some of the most egregious manipulations of county caucuses . . . in recent American politics. And I don't like people stealing elections, and I don't like people doing things that are fraudulent, and I don't like people breaking the law. And we've seen a good part of that in Michigan."

Complaints by Robertson and his allies of illegal tactics were severely undermined here by acknowledgment that by far the most questionable tactics of the caucuses were at a meeting in Elmer's Steak Pit Hall near Dearborn, where the pro-Kemp-Robertson chairman, Perry Christy, made Bush delegates wait outside in the 10-degree weather. Christy arbitrarily decertified eight Bush delegates in order to gain a majority, and then unilaterally declared himself and a colleague to be voting members of the gathering when it became clear he needed two more votes to win procedural disputes.

The event at Elmer's, which received extensive media coverage, was so embarrassing that Kemp officials today sent a letter of contrition to the vice president. "It was appalling and totally unrepresentative of Jack Kemp. We apologize to the vice president, his son {Marvin, who was there} and to the hundreds of people who got stuck out in the snow," Dick Minard, Kemp's Michigan manager, said.

The Bush forces were not subtle in their attack on Christy: "Perry Christy is an only child which goes to show that his parents learned from experience," L. Brooks Patterson, the Bush cochairman, said.

The outcome of the conventions remains in dispute, but new and potentially lethal cracks began to surface in the Robertson-Kemp coalition. If these conflicts break the conservative alliance, Bush would almost be guaranteed a plurality of the delegates when the Michigan GOP holds its statewide convention in two weeks.

"We had some indications that the local arrangements between Kemp and Robertson might not have been honored," Charles Black, Kemp's national campaign manager, said of a night marked by angry walkouts, rump conventions and the attempt by Christy to force members of the news media to pay as much as $850 to cover the event.

Black said preliminary reports suggested that Robertson forces may not have lived up to prearranged deals for state convention delegates. "A coalition that cannot produce might not be as valuable as it used to be," he said.

Kemp's state leader, Dick Minard, said, "All we are saying is we are looking at our options."

One key tactic still available to Robertson to block Bush is to use his control over the 101-member Michigan Republican central committee to set up a credentials process that would bar the Bush delegates from participation in the state convention. Robertson's control, however, is razor-thin and a collapse of the alliance with Kemp would shift central committee power back to party regulars committed to Bush.

The National Election Service said Bush won at least 919 delegates to the state convention, 57 percent; while Robertson got 360 or 22 percent; Kemp 274 or 17 percent, and the rest split among other candidates or have not yet been counted.

Even with some delegates yet to be counted, NES gave Bush more than enough to control a majority of the 1,805 delegates to the state convention, which, in turn, will pick the 77 delegates to the August national convention. In addition, an undetermined number of Kemp delegates are defectors from the Kemp-Robertson alliance and are committed to vote at the state convention for a deal giving Bush first place and Kemp the second spot. The NES numbers do not include results from the 33 rump conventions, five of which were held by Bush, the rest by Robertson-Kemp supporters.

Rich Bond, Bush's national political director, and Mary Matalin, the campaign's Midwest director, said Bush won a total of 1036 delegates, including those selected at five Bush-initiated rump meetings. The Bush delegates will be bolstered, they said, by 140 Kemp delegates who defected from the conservative alliance and will vote with the vice president. The Robertson-Kemp coalition won 584, by Matalin's count, with the remainder split among other candidates.

These findings contrasted sharply with the results claimed by Ben Waldman, Robertson's deputy press spokesman, and David Walters, Robertson's Michigan manager, which included results from the 28 pro-Robertson rump conventions: 848 for Robertson, 466 for Kemp, 448 for Bush, and the rest scattered. Officials of the Robertson campaign said they believe two-thirds of the Kemp delegates will support the Kemp-Robertson coalition, giving it a majority of the 1,805 and that one-third will defect to a separate alliance with Bush.