One of the main things I enjoy about MO and MSE is the chance to solve somewhat difficult problems. On a couple of occasions, I've found proofs of results (posed as questions mostly on MSE) which seem original and not entirely trivial. The examples I'm most fond of are short but tricky, not in any way revolutionary, but have a satisfying "aha!" at a crucial point.
Disappointingly, these questions seem to always disappear into the mists of time. While my answers to "soft questions" often get enormously upvoted and see lots of views, those answers to serious questions which actually have some original intellectual content often go largely unnoticed. Once they fall off the top of the question list, they essentially are never seen again.
Perhaps this fate is inevitable, but it seems a shame to let a cool trick die. This has got me wondering whether to try to publish any of these. I'm not a professional mathematician, so this likely would not materially benefit me in any way. The only goal would be to put the results out in a place where they might be of use. The options I know of are:
- Drop it on arxiv.
- Publish in the usual way, in a topical journal.
- Publish a note in an expository journal like American Mathematical Monthly.
- Submit as a problem to a journal that publishes problems.
I've published in the Monthly before, and it seems a good place for small results of broad interest. However, it's not without cost (submitting takes work), it's not totally clear that results published in the Monthly "do any more good" than those rotting in the bowels of MSE, and moreover not all minor results are of sufficiently broad interest for an expository journal to be a viable option anyway.
So my questions are:
- What's the best way(s) (in the sense of "being maximally useful to those who may find them interesting") to disseminate minor mathematical results?
- Is it worth the effort? Or are all minor discoveries bound to languish in obscurity until such time that they are rediscovered and applied to a major result?
- Are there any good rules of thumb to determine what minor results are worthy of dissemination (if any)?
A couple afterthoughts:
- I've noticed stackoverflow doesn't have the same problem with questions not having any later use, presumably because programming questions are more likely to be shared by many people.
- I can give examples of the minor results that inspired this question if requested.
- I think this question should be community wiki, but apparently I'm not allowed to designate it as such myself.
Best Answer
I'm in somewhat similar position too. My attitude is that most time people know why they are asking for some particular result so it is their headache to convert it into "public good" and to "spread the word", so to say. I have some other fish to fry (including thinking of other math. problems even from the same MO). Sometimes I like the result so much that I even present it on a seminar or a colloquium talk. However I'm just too lazy to go beyond that and to make a formal publication out of it (that always comes at a cost, not just if you submit to Monthly).
But that is about me. Your attitudes may be quite different. So, let me try to answer your formal questions now.
1) You've done the most essential part already: you communicated the information to a person who really needed it and openly declared so. True, there may be some other people who might find it interesting, but there are many other interesting things for them too, so no efficient directed communication is usually expected or even possible if you just have a funny trick to share. You can always throw a few more drops into the huge river called "mathematical literature" and there is a non-zero chance that somebody will swallow one of them when drinking his daily cup (bucket, barrel, cistern, perhaps even "pond" for some really top guys, still never the whole river or anywhere close to it) of information but the only sure way to attract reasonably wide attention to your trick is to use it to solve some known open problem.
2) No, it isn't worth the effort. But so isn't the life unless you are one of very few people who really make a noticeable change in this world for the better. Still we all are living just for the fun of it, so if you feel like writing an article, such considerations shouldn't stop you. As I said, I prefer thinking to writing any time I have the choice, but there are many people who love writing so, if you are one of them, just go ahead and write.
3) There are several contradictory ones. So, just rely on your taste and common sense and proceed by trial and error. Don't get surprised though if it turns out that the ideas of other people about what is interesting and what is not don't coincide with yours more often than not. If you really want to write for them, try to understand their point of view instead of getting angry and frustrated. Snobbish remarks of the type "this is not research level" count as "a point of view" only if the person making them can do what you've done with his little toe on the left foot alone, but there are justified explanations of why some things are not worth publishing (the most typical one is "Lemma 5.1 in this textbook covers your approach in a much more general setting"). Warning: what is new and exciting for you may be a boring routine for humanity; after all it is not easy to notice something that seven billion other people overlooked. So, you'd better have some indication of an outside interest to your idea before trying to spread it unless you want to end up in the same position as I when, being a pupil in a math. school, I decided to accurately write some complicated proof by induction of the fact that the Euler function $\varphi(n)$ is even for every $n>2$ to impress my classmates with it. The end of that story was most embarrassing, of course.