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I learn math on my own. And I sometimes end up generalizing theorems. I do not claim that these generalizations are ground-breaking. However, I feel these generalizations are not entirely obvious at first-sight.

Could I publish a paper on such a generalization?

Making this more relevant to the community at large, how do you know when your "new" ideas are paper-worthy? Should one pursue research directed only by external trends? As in, if I were to publish a paper, should I only look for current areas of research in order to conduct research that would be relevant to academia today?

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    Talk to someone involved in research-level mathematics (can be even a student). You will get some kind of feedback (it is hard to say anything in general). Especially as even if the finding is interesting, its clear presentation counts as well. – Piotr Migdal Nov 3 '13 at 13:39
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Complementary to Suresh's and Peter's more comprehensive answers that you should definitely take into account: (+1 to both)

Use ArXiv; ie. publish it yourself. Go ahead and write your findings down and put them in public. This will be a good exercise as :

  1. you will be covered for plagiarism etc. and you 'll be also able to refer other people's attention to it. It will be immensely easier to attract people attention to something tangible than just referring to "some idea you think it is great". As Torvalds said : "Talk is cheap. Show me the code." (or Maths in your case).
  2. people you do not know, can actually find you; or even cite you for that matter. I know a lot of people who regularly read ArXiv papers to keep up to speed with the bleeding edge of stuff. You might be lucky and really get some attention from people that actually care for your work.
  3. you will see for yourself if what you wrote can be formulated in a research paper and it doesn't come across as some ''back of the envelope'' calculations. You might even identify where feedback from a collaborator would be helpful.

If you think you are up to something good, put it up there. Worse case scenario: nobody bothers and you never know if you were right or wrong.

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    If I understand correctly, the author is not affiliated with a university -- it might be difficult to get endorsed to publish on arXiv. – Federico Poloni Nov 3 '13 at 23:07
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    +1 for self-publishing. A word of caution - send yourself a copy of your paper in the mail (real, letter-in-an-envelope snail-mail) to get it postmarked. Do not open the envelope. Keep it safe. Only once you receive the letter back should you contact potential endorsees with a copy of your manuscript. Most people are honest, but better safe than sorry. – Moriarty Nov 3 '13 at 23:38
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    @Moriarty: Of course you can do this if you want. But I think this advice feeds an unhealthy thought pattern that seems to be common among amateur mathematicians: that their work is potentially groundbreaking and the mainstream mathematical community would love to steal it. Unfortunately, what is much more common is that the work is wrong, or already well-known, or so confused as not to be comprehensible to a practicing mathematician... – Nate Eldredge Nov 4 '13 at 1:09
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    And if an amateur has an inflated opinion of their work, or takes paranoid steps to protect it, that tends to be off-putting to a professional and makes them even less interested in helping the amateur to develop their work. – Nate Eldredge Nov 4 '13 at 1:09
  • @NateEldredge If ensuring ownership of your work is conducive to an inflated ego, then that is simply irrational. Obviously such precautions are only reasonable if you are sending people you do not have reason to trust a copy of your work. – Moriarty Nov 4 '13 at 2:12
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This is a very hard question in general, and is defined by a combination of absolute standards and community opinions. It's very hard to answer your specific question ("can I publish these results") and I don't think this community is the right forum to even ask that question.

But your second question is a good one. There are some things to keep in mind:

  • make sure your ideas are indeed new. It's easy when working independently to be aware of the published literature, but not be aware of the large cloud of "folklore" knowledge that floats in the air of a community. You'll get a paper rejected because something is "well known", even though you can't find a specific reference and no one provides one. To figure this out, it would help to approach an expert in the field, or at least someone you trust.

  • What kind of research you pursue is a complex combination of external trends and your interests. If you go too much towards external trends you might not find much pleasure in it. If you go too much towards personal interests (unless you have impeccable taste) you might find yourself isolated. It's all about balance. Similarly, while looking at current areas of research will tell you what's likely to be publishable, that's not the same thing as what's interesting or useful. Again, having some contact with people in the community might help a bit.

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The only way to "know" if your material is publishable is to know the literature on the subject. I used "know" because all publications go through per review and that is the final hurdle your paper must pass before the answer is given. So the question really becomes what must be done to pass peer review?

So you need to know your subject by finding and reading all relevant literature. An interesting personal observation is that it is easy to think you know more than you really do if you are not familiar with a specific field. Getting yourself up to date is hard work. Apart from the research literature, there is of course need to master the basics. As editor and reviewer I see many manuscripts presenting measurements that in themselves might be good but where interpretations are shallow and sometimes trivial because they have missed relevant research. Remember that journal rejection rates, although they vary, can be from below 50% to up to 90% for the more prestigious journals.

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Even professional scientists cannot know if their last idea is original and new (and so, if it is worth publishing).

Today science is too big, too complex and too separated in different specialties to allow one person to have a general overview (this is the "Big Science" issue).

However, there are people who are specialist of evaluating scientific contributions. There are referee of Journals.

So my answer would be to try to be published. Even if the paper is not accepted, it is the opportunity to have feedbacks, references and to see what your idea deserves.

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