Along a rough, scrubby stretch of Interstate 25, heading south between Santa Fe and Albuquerque, someone scrawled on an old, rutted billboard: Viva Bush.

It is the most obvious sign of support for President Bush in these parts. At least in the corridors of this desert state's most populous cities, in and around Albuquerque and Santa Fe, Bush's Democratic challenger, John F. Kerry, is easily winning the sign campaign. Unidos con Kerry posters dot the living room windows of the short, coral-pink houses that line city streets, and Kerry-Edwards bumper stickers outnumber the Bush-Cheney stickers by what looks like 10 to 1.

But the sign war is just a small, symbolic part of the ground war that the two campaigns have been waging in this swing state, where Al Gore beat Bush by 366 votes, or less than 1 percent of the vote, in 2000. No one is claiming a win in the fight for New Mexico. Despite the Democrats' showier presence, punctuated with celebrity appearances for Kerry such as singer Carole King's three-day campaign swing through the state last week, the latest polls show the race a dead heat. The two parties are fighting for every vote, and a big part of the fight is directed, more than ever before, on winning over Hispanic voters.

Republicans have their "Viva Bush" campaign, an effort in which field campaigners in all 33 counties try to engage Hispanic voters on the issues. They have also been deluging the airwaves with ads since March, in Spanish and English. The Democrats, whose ad blitz is more recent, have field offices geared for Hispanic volunteers to court Hispanic voters, with bilingual door-to-door canvassers and phone bankers.

"Everything that this campaign is doing in New Mexico takes the Hispanic vote into account," said Danny Diaz, a spokesman for the Bush-Cheney campaign.

The reason is obvious. New Mexico has the highest proportion of Hispanics in the country -- 43 percent of its 1.8 million residents, or about 30 percent of the vote.

But Hispanics here are a tricky voting bloc, if they can even be called a voting bloc. Unlike in California, where the majority of Hispanics, or Latinos, are fairly recent immigrants, two-thirds of New Mexico's Hispanic population traces its heritage to the Spanish explorers who ventured to the New World before the Mayflower landed on Plymouth Rock. So while Democrats hold a distinct advantage among Hispanics here -- with 58 percent supporting Kerry, according to the latest Albuquerque Journal poll -- the Hispanic vote is swingy, fickle.

New Mexico's Hispanic voters do not call themselves "Latino," do not consider immigration a pressing issue and tend to side with the GOP on social issues such as abortion and same-sex marriage. A great majority of them were born in the United States. Both parties believe that Democrats will win a majority of the Hispanic vote. But both sides also believe that if Republicans win 40 percent of the Hispanic vote -- 5 percent more than the 35 percent Bush garnered in 2000 -- the president will win the state and, hence, the country.

New Mexico is well aware of its history of picking winners. Since it became a state in 1912, it has picked the winning candidate for president every time, with two exceptions: Gerald R. Ford in 1976, and Gore in 2000 (though he won the popular vote). The Hispanic vote is a crucial part of the winning equation.

New Mexico's popular Democratic governor, Bill Richardson, a bilingual Hispanic, calls the 40 percent mark "crucial." "Our objective is to maximize turnout and bring the new voters to the polls," said Richardson, who last week taped television commercials in Spanish and English to help Kerry. Voter registration rolls have swelled by 106,000 -- and counting. When all are counted, the total could be 140,000, said Brian Sanderoff, an independent pollster here. "We've isolated all the Hispanic surnames in our list of 106,000 and it comes out to 30.1 percent -- the same percentage as in the overall voting population," he said. Democrats have a registration advantage of 52 to 32 percent, or 1.6 to 1. But with Democrats less loyal than their GOP counterparts, Sanderoff said the governor is right to say Kerry needs a majority of the Hispanic vote to win the state.

Early voting began Saturday, with county clerks throughout the state describing turnout as heavy.

Republicans say they have more than 14,000 volunteers, have knocked on 40,000 doors, made almost half a million phone calls and registered 50,000 GOP voters for their effort. The Democrats will not reveal their numbers, but say they have far outdistanced the Republicans in their ground-troop efforts.

"The reason we feel confident is not just because we have the better ground campaign," said Ruben Pulido Jr., spokesman for the Kerry campaign, "but it's also because we are on the right side of the issues. Hispanics in this state care about the same issues as all other voters -- health care, jobs, education. And on those issues, they see that the president has not delivered. Half of all Hispanics do not have health insurance, for example."

One of the Democrats' secret weapons in the ground war is Tony Garcia, a 24-year veteran of the Marines who retired in June and is a paid staff member for the Kerry campaign. He had not voted for 24 years and realized his opposition to the war in Iraq only once he returned, he said.

"I realized that this war is wrong and that we'd better get Bush out of office," said Garcia, who adorns his desk at the Kerry campaign office with medals he won for his combat service in the Persian Gulf, Afghanistan and Iraq wars. Part of the flag that was taken when Marines toppled the statue of Saddam Hussein is draped over his desk.

"A lot of Hispanics are veterans, and I talk to them about my experience and why it is important to vote for Kerry," Garcia said.

But Republicans have their Hispanic veterans, too. Gabrielle Escovedo, wearing a "Viva Bush" button as she strolled through the J.C. Penney store in Albuquerque, said her son was in the war in Afghanistan. "He says it is important to support our troops through our president," she said. "So I tell everyone who asks me, in Spanish or English, to vote for Bush."

White House counsel Alberto R. Gonzales, at lectern, with presidential nephew George P. Bush, left, SBA Administrator Hector Barreto and former U.S. treasurer Rosario Marin, campaigns for the GOP in Las Cruces, N.M.