Q:

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.)  

A:

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.

A:

The answer is "yes", but many advisors may respond differently to this. I think it depends much more on the advisor than on any established "academic norm":

Advisor might not be happy that you are using up your time for "trivial pursuits" on the advisor's grant money when you should be doing "real work toward your thesis". (For the record, I would consider this as a bad reason to be unhappy with your students). Often, if you can ensure that the side project doesn't take up that much time, you can mitigate this risk a little.
Advisor might be happy that you are independently pursuing projects.
Advisor might not want a part of this paper simply because of lack of time/energy but otherwise be happy that you're doing it. This is pretty common, for example, in the case of class projects in which you end up with something that's actually pretty significant and the class professor would like you to help publish it.
Advisor might not want you to "spread yourself out too thin" later in your PhD career. I've been advised to be careful about coming up and getting involved with too many "one-shot" ideas that will never get developed and don't help your overall image. So, for example, if you're in the area of "program analysis and testing" (for example), publishing a one-off paper in a venue (maybe "distributed computing") that you don't keep up with and won't be remembered in will result in a forgettable, low-impact paper. I think this is more of a risk when students are thinking "of things to work on" and aren't really focused or don't have a good idea of a research thread to develop. This is also more of a risk if your academic profile isn't very focused. For example, if you have a bunch of disconnected topics you're more in danger than if you have only one or two side projects in a whole field of papers on your main interest.

As a note beneath this one, I think most advisors are mostly concerned that their students are "too distracted". It's easy to do that as a Ph.D ("I can explore anything I want!") - so advisors by nature of their jobs need to make sure that the thesis ends up having focus. Having ideas is a good thing, but putting those ideas into papers is a lot of work. Just to help put a bit of perspective here.

I am generally in favor though of at least letting your advisor know of "side projects" that you're involved in, even class projects, because often they will want to find ways of integrating that into some research work that you can be doing. A good advisor might see a connection between your combinatorial geometry "side project" and your main thesis research, for instance. It's also a good idea, in general, to have a tiny amount of breadth across one or more areas as well. It not only generates good ideas, but helps you keep perspective.

A:

Rather than repeat what others have mentioned in their answers, I'd rather touch upon another related issue:
In some cases not involving your supervisor also means not involving the institute that you are part of.
Apart from the issue of attribution, depending on the specifics this may have legal ramifications, especially if you are receiving funds from a grant. Theoretically, if you have used any resources supplied by your institute, including existing ideas, computing power or your own paid work hours, you may not have sole ownership of the resulting IP. Things may get even more complicated if your co-author is colaborating with a different entity.
You may not need to do anything, you may need to get a waiver of IP rights from your institute or you may have to add something along the lines of a "This work was supported by..." snippet. If I were you, I would discuss this with my supervisor, even if only to clarify any such issues...

