I (independently) solved a fellow student's research problem. I want to publish it. What should I do? A fellow student and I are part of the same graduate student research group (in applied mathematics), under the same PhD supervisor. The (relatively new) student came to the group with an idea for a thesis, and we helped him come up with a different research idea that more suits our research interests, but still ties to his original goal.

The current formulation of his research problem is mostly due to input from me and our mutual supervisor. As in the model is based on our suggestions across multiple meetings.

Being interested in this student's research problem as well (or at least my formulation of it), some months ago I offered that we collaborate on it. Not much came out of it, mostly, I think, because this student is still very inexperienced, and his mathematical maturity is not so high.

Some time ago I came up with a different partial solution for the aforementioned research problem. As part of our (failed) collaboration I asked him several times to update his simulations to consider this solution, but he seemed to be against the idea, believing that the original formulation is better. Our meetings mostly consisted of arguing about which model is better and me trying to teach him some mathematical concepts he can use to analyze his work.

A few days ago I have (independently of the student or our supervisor) come up with a solution which is better, and completely different from what the student has been trying to analyze. The solution invalidates a lot of the student's work up until now (he has been working on this problem for a few months), since it is simpler and more effective.

I want to write up this solution in a paper and publish it. I already have basically all of the results written down (except one lower bound which I am still working on), and a nearly complete first draft. 

Here's the problem. Do I include this student as a co-author? I am conflicted, because this paper really is my own original work. However, it is based on the research problem which our group "handed over" to this student. I do not wish to put him in the unfortunate situation of having months' worth of work invalidated, and everyone in our group views the problem I have solved as "his" problem. However, it is also true that this paper consists entirely of my own research. 

Ideally in this case I would approach the student and ask him to collaborate with me on the paper, as in, refine some of the results, or simplify some of the proofs. However, based on many conversations with him, I believe that the student is too inexperienced to do this, and is not familiar with the main mathematical tools I have used to prove the paper's result.

How should I proceed?

Some more details: I should mention that I have no personal qualms about sharing authorship. I am mostly worried about two things:


whether it is the most ethical thing to do;
I feel that if I include him as a co-author, he will want to make a contribution to the paper. But I want to have complete creative control on the paper, and feel that sharing creative control with him is going to slow me down.
There is a larger issue at hand that you seem to be missing.

Yes, I do believe that you can publish your own work, and probably independently if neither your advisor nor your colleague helped. And I understand that there are some field-dependent conventions in mathematics that would be best to discuss with your advisor.

That being said, I believe that, being a senior and more experienced student, you should not have allowed the situation to have reached this point. That is to say you should not have scooped your colleague in the first place. You offered your help to collaborate with your colleague and ended up quite literally stealing his project and are now thinking about publishing it because he was not able to work as fast or as efficiently as you (because he is inexperienced, as you said). This is something that should have been discussed with your advisor at a much earlier stage.

You should understand that this will probably not be the last time that you can benefit from collaboration, and you don't want to be known as the person who steals other people's projects. Healthy collaboration is an intrinsic part of academia (and society, in general). You are expected to be trusted to review other people's papers without scooping them, and in the future you will likely advise inexperienced students in their own projects, many of which have been suggested by you, and that you could solve with much less effort. More importantly, you will be partly responsible their success. 

Speaking from your advisor's standpoint, I would consider your action harmful to the group's health. While I generally applaud collaboration between students, in my group, projects and milestones are set with the understanding that students can benefit from them, and that they can help them progress in their more ambitious goals (i.e., a dissertation). This is particularly important for inexperienced students. Should there be a major change of directions (such as a student unable to make progress and someone else benefiting from it), I would like it to be run through me (or other senior member of the lab) first so that it can be arranged for nobody to be in a losing situation. In your case, your colleague is in a losing situation, his name going in the paper or not.

The moment you noticed a solution to the problem, you should have informed your colleague (and possibly your advisor) and both of you should have worked on it, together.