WEST PALM BEACH, Fla. -- Sen. Phil Gramm turned up here the other day for a breakfast with 150 Florida Republicans after similar stops in Miami, Tarpon Springs, St. Petersburg, Tampa, Orlando, Jacksonville, Sarasota, Fort Myers and Coral Springs.
Lamar Alexander spent a couple of days in the state last week, and Senate Majority Leader Bob Dole popped in, as he does most weekends these days.
All this attention from three prominent presidential candidates, and an occasional foray by Patrick J. Buchanan, suggests that something big is in the wind here a full five months before Florida Republicans will vote in a primary to choose delegates to the party's nominating convention.
In fact, all this is directed at a straw poll -- but in this case one the politicians consider the Big Enchilada. It will be held Nov. 18 when about 3,500 delegates gather in Orlando.
The event is being run by Jeb Bush, who lost a gubernatorial election last fall but clearly intends to run again. His intention, he said, is to energize the party with new supporters. To that extent, he has succeeded. More than 11,000 Florida Republicans threw their names into the hat for the 2,700 delegate seats awarded in 67 counties last month.
The candidates responded with a rush to win their favor. "We've achieved our objective of putting our party activists in a privileged position," Mr. Bush said.
What makes the Florida poll so attractive to the candidates is, of course, the prospect of media attention -- both nationally and in a critically important vote-rich state -- and the likelihood that the results will be taken more seriously than the stakes would justify. At the last such convention in 1987, 267 media credentials were issued.
Or, as Mr. Dole said recently, "Whoever does well is going to have what we call momentum."
History suggests, however, that there may be as many hazards as rewards here.
In 1979, John B. Connally startled reporters by predicting that he would run at least even with Ronald Reagan and probably beat him. When Mr. Reagan won with 36 percent to 27 percent for Mr. Connally and 21 percent for a lightly regarded contender, George Bush, much of the air went out of Mr. Connally's balloon.
In 1987 Vice President Bush was such an overwhelming favorite that his two prime rivals, Mr. Dole and Jack Kemp, pulled out of the competition. Mr. Bush won with 57 percent, but some of the luster was taken off when Pat Robertson wowed the audience and came away with 38 percent.
In this case, the candidates themselves seem to be raising expectations. Mr. Dole has said he won't be the front-runner for the nomination if he loses here.
Mr. Gramm, who already has won nine straw polls that received less attention, predicted last spring that he would win here.
And Mr. Alexander, the former governor of Tennessee, says he will prove in this straw vote that he should be taken seriously.
No one knows how the delegates will vote. Mr. Dole has an advantage because about 800 of the 3,500 delegates are party officials, officeholders, big contributors or people chosen by party officials -- in short, the establishment Republicans with whom he is leading everywhere at this stage.
And the Dole campaign is credited with having a sophisticated organization that benefits from the experience of, among others, Richard Pinsky, the political director who worked for the Robertson campaign eight years ago.
As he demonstrated here, Mr. Gramm strikes a responsive chord in many Florida Republicans with his focus on tax cuts.
"I like what he wants to do about taxes," said Todd Sipowski, a young insurance agent from Jupiter. "I make $30,000 a year, and they take $10,000 and leave me $20,000. And I've got a wife and two kids and another on the way."
But the Texas senator's highest card may be his reputation for uncompromising conservatism. His audience cheered when he talked about "people who do the work and pay the taxes and pull the wagon in America" and declared: "I wouldn't want the government we have in Washington, D.C., even if it were free."
Mr. Gramm contrasts himself with Mr. Dole by suggesting that the majority leader has made too many compromises.
"I know who I am," he said. "I know what I believe in. I'm not afraid to stand up on an issue where I know I'm right even if it's not popular."
He said in an interview, "My record will sustain my rhetoric."
Mr. Alexander is making an intensive effort here in what he sees as a chance to rise out of the second tier of candidates to which he has been relegated on the basis of national polls. Tre Evers, his campaign director here, said the Tennessee Republican has made about 20 visits to Florida this year and met with delegates from about 15 counties.
But strategists estimate that at least 40 percent of the delegates are uncommitted, many because they've decided they like the idea of being wined, breakfasted and wooed by the candidates. Said the Dole campaign's Mr. Pinsky: "There's definitely movement from the committed column to the uncommitted column."
On the face of it, the prize here -- a day or so of glory on the evening news -- wouldn't seem worth the shoe leather or the money candidates will spend in the next month on mailings and telephone canvasses. But, as Bob Dole said, on Nov. 18 someone "is going to have what we call momentum."