publications - Can two PhD students publish without involving their supervisors?

I am a theoretical computer science PhD student. My childhood friend (my present roommate) is in the 3rd year of his pure mathematics PhD program. Very recently we both worked on a problem on combinatorial geometry and got some interesting results to publish. However both of our dissertation topics are way different from this work, so we don't want to involve our advisors in this matter. This was kind of our joint fun project. We don't know whether it's academically unethical to publish paper(s) as PhD students without including our advisor(s).

(Note: We have nothing to lose even if they reject our paper right away, but we don't want the editor of the journal to mail the chair about this matter. Maybe I'm thinking too much because I've never done this kind of thing before.)

Answer

There is nothing unethical about publishing something like what you suggest. Personally I would be happy and encourage a student of mine if that happened. So from a formal side you need not to worry. I can add that authorship, or contributorship, does not include adding names to a paper if they have not contributed anything (or enough; see posts on authorship on this, Academia.sx, site). I would, however, be open about it with your advisor. I assume you have a good working relationship with him/her? The only thing that could complicate things would be if you are in a bad working relationship with your advisor or if your system is very hierarchical and not open to initiatives. Clearly only you can assess this. But, I do not want you to over-emphasize these "risks". If you get stuff published on your own and in a field that is not directly within your topic, it will only be viewed as a positive in your resume when applying for, for example, post-doctoral positions.

As for risking rejection, I suggest you have someone whose views you trust to read and comment on the paper. Having someone independent look at the work is always good to work out details that can otherwise distract reviewers. This is always a good idea so it is not unique to your case.