I'm an undergraduate currently doing summer research with an REU program. I feel that the research we've done this summer is publishable (although what do I know), and I'd like to do that, especially with grad school applications coming up. Is there a polite, yet direct way (or is it uncouth) to ask my advisor if we are going to publish this paper?
Hi, do you think we can get my work so far published as an article?
If the answer is no:
So, what else do you think we would have to do to get it publishable?
Professors and supervisors are human beings. They won't eat you if you make the wrong question, or use a slightly unproper tone. Don't worry specially when you are in a very low risk situation, and you know each other.
(Once you get the go, don't forget to clarify and define authorship).