The most important thing is your recommendations. If you have three professors who went to a top-tier school, and who will give you a good recommendation, then

The most important thing is your recommendations. If you have three professors who went to a top-tier school, and who will give you a good recommendation, then you should get in. So, how to get a good recommendation. Just ask your professors, would you give me a recommendation, and would you expect me to get in. If they are actually going to give you a good recommendation, they will tell you that they definitely expect you to get in. If they are sort of noncommittal, you are probably going to get a mediocre recommendation. Ask them what you would need to do to be a great candidate. Then do that. Publication helps of course, especially in convincing your professors to give you a good rec. If you don't have any publications or ideas on how to get some, you might try a REU.