How to figure out timing for working on and submitting independent research? I am a PhD student in mathematics and have an idea for a general theory, which in a sense is an intermediate case between two opposite ends of well-understood theories. This intermediate case has gained little attention in the literature.

I came up with this idea for a more general theory by studying an explicit example in this intermediate case. The theory includes the two well-understood opposite ends in a trivial way.

I have checked with my supervisor, and he said it was a good idea and "I should write up the details". I have done so, but it has been sitting like this for almost two years. It's very tangential to the research of my supervisor and, as far as I can tell from trying to get his input, he is more interested in this particular example and not in the theory that should be built around it.

I read in "How to protect your unpublished ideas?" that the best thing would be to put this idea on the arXiv and try to submit it as a paper.

I obviously can't ask you here whether my idea is really good. But other than my supervisor, there is no faculty with this area of expertise at my university. I also don't know any other professors well enough to simply write an email asking for an opinion.

Developing the theory in all generality would be time consuming. So I'm wondering which of the three options would be best:


Submit a paper now, co-authored with my supervisor, using just my example and hinting at the more general theory.
Working out the details for some other cases and try to publish alone.
Wait until I have more time to focus on this project (after my PhD).
Put a preliminary version on the arXiv, running the risk that it contains some errors (errors on the order of magnitude of forgetting additional hypotheses, I know it works in a few examples).


In any case, I would confer with my supervisor to see whether it would be okay for me to try to publish something by myself.

I should also say that, for my taste, I'm a little short on papers (I have two short papers in B-grade journals) and I'm looking for a PostDoc soon.
I can not give you a fully fledged answer at all, but some things to take into consideration:

Where do you want to place your article? This determines the amount of time you will have to spend on it. This has to be in-line with your idea about being short on papers (because it might not help the issue if noone reads the journal).
I personally would find it a tad rude to publish alone, if the idea has spawned from something connected to your work for your insitute (and might also not be allowed depending on your contract).
If you are not pleased with option 1 with respect to your ideas about the paper's benefit for you I would go with 3. You could also think about finding a PostDoc position which includes the work on this subject. If you do this intelligently, you could also use this as a selling point for yourself. Ideally this also would be the general idea of a perfect PostDoc position, allowing to develop your own ideas and research. You could try doing so with a research stipend while going abroad.