Cain has benefited not only from his debate performances, but also significant media attention after winning the Florida Republican Party’s straw poll on Saturday.

Perry now garners 19 percent, a drop of 10 percentage points from a month ago. That puts Romney back in the top spot with the support of 23 percent. Last month Romney was at 22 percent.

Newt Gingrich recovered some ground and now stands at 11 percent. Ron Paul receives the backing of 6 percent now compared to 8 percent before the September debates.

Bachmann registers 3 percent support, down from 8 percent in late August and a high of 15 percent in July.

The nomination preference question included only the names of announced candidates.

The order is similar when GOP primary voters were asked which candidate they have the most in common with: Perry tops that list at 17 percent, followed by Cain at 14 percent, Romney at 12 percent, Bachmann at 10 percent, Gingrich at 7 percent and Paul at 6 percent.

Republicans are much happier with the GOP field these days. The number saying they are at least somewhat impressed with their slate of candidates has increased 19 points -- from 44 percent in April to 63 percent now.

After much speculation, New Jersey Gov. Chris Christie said again Tuesday he would not run for president this year. The poll found that by a 7-point margin Republicans thought Christie should run and by a 4-point margin Tea Party folks liked the idea.

What about the other side of the 2012 ballot? By a 66-27 percent margin, Democratic primary voters prefer Obama as their nominee over his former rival, Sec. of State Hillary Clinton. And another 3 percent volunteer they would like someone else altogether.

Hypothetical, Early Matchups

Obama continues to hold a slim edge over Romney and Perry in hypothetical matchups among all voters, but not among those highly interested in this election. That’s driven by relatively higher levels of interest among Republicans as their party picks a candidate and holds debates.

Among all registered voters Obama tops Romney by 3 points, which is within the poll’s margin of sampling error. In July, Obama had a 6-point edge and a year ago it was 1-point. The strength of party support is similar for each, as 85 percent of Democrats back Obama and 83 percent of Republicans support Romney.

Romney tops Obama among independents by a slim 2 points.

In a head-to-head matchup with Perry, Obama bests him by 8 points among all voters, and by 6 points among independents.

Overall, more voters doubt that Obama will be re-elected president than think he will be (by 50-to-40 percent). In May, 57 percent thought Obama would be re-elected. That was soon after the killing of Usama bin Laden. By a better than 2-to-1 margin Democrats think Obama will be re-elected.

While a 56-percent majority sees Obama’s strategy for re-election as bringing people together “with a hopeful message,” a third of voters (32 percent) think his strategy is to drive people apart “with a partisan message.”

Don’t Bother Lunching with Trump

Since Donald Trump toyed with running for president earlier this year, several Republican candidates have made a point of meeting with him. Even so, hardly any voters -- 6 percent -- say they would be more likely to vote for a candidate endorsed by Trump. In fact, by a 5-to-1 margin a Trump endorsement is more likely to have a negative impact.

Almost twice as many Republicans say a Trump endorsement would make them less likely to vote for a candidate than say it would encourage their support.

The Fox News poll is based on landline and cell phone interviews with 925 randomly chosen registered voters nationwide and was conducted under the joint direction of Anderson Robbins Research (D) and Shaw & Company Research (R) from September 25 to September 27. For the total sample, it has a margin of sampling error of plus or minus 3 percentage points.