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I read in "How to protect your unpublished ideas?"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.