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Question. How does one know that a theorem is strong enough to publish?

Basically, I have laid out a framework in which many theorems may be proven. I'm only 18 and therefore lack knowledge of whether this framework and the theorems sprouting from it are trivial along with the theorems. What is a good indicator that work is good enough to be published?

An example of a theorem I have proved is;

Given a (non-constant) meromorphic function $f$ there exists at least one continuous loop over the extended complex plane, $\varphi$, such that $f\varphi :\mathbb{R}\rightarrow \mathbb{R}$ (bijective).

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This is false. Take $f(z) = e^z$. –  Qiaochu Yuan Aug 5 '11 at 20:38
If you're 18, then presumably you'll be starting college in the fall. In that case, show your math prof your work. –  Ben Crowell Aug 5 '11 at 20:42
$e^{z}$ is not a counter example to the theorem if you consider the extended complex plane. –  Harry Barber Aug 5 '11 at 20:50
@Harry: "meromorphic function" generally means meromorphic on $\mathbb{C}$. A meromorphic function on the extended complex plane is a rational function. The result is still false if $f$ is constant, and if $f$ is non-constant then it is surjective because $\mathbb{C}$ is algebraically closed; in particular, it's surjective onto $\mathbb{R}$. Using the fact that a rational function defines a branched cover it should be straightforward to lift $\mathbb{R}$ to the desired path. –  Qiaochu Yuan Aug 5 '11 at 20:54
@Harry: the idea is that $f$ has a local inverse around any point $a$ where $f'(a) \neq 0$ by the inverse function theorem, so you can locally take the preimage of parts of $\mathbb{R}$ to get parts of a path mapping to $\mathbb{R}$. There is no problem with doing this unless $f'(a) = 0$ at a point where $f(a)$ is real. At such points there will be branching in the preimages (parts of lines will split or combine), but in any case this only occurs at finitely many points and there should be no obstruction to picking a consistent choice of branches. –  Qiaochu Yuan Aug 5 '11 at 21:13
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2 Answers

Regarding the general question, it seems to me a slightly more pressing question is whether the result is well-known or not ("known" is trickier; various things were known at some point and forgotten to various extents, and it may not be a bad idea to republish such things), or otherwise easy enough to deduce using known techniques. The attitude I think is appropriate here is one of humility. Just consider the fact that smart people have been doing mathematics for thousands of years, and in this particular case smart people have been doing complex analysis for centuries. For relatively old fields all of the easy results are likely to have been proven already, or at least that that should be the default assumption. To assume anything else seems to me a little arrogant.

For example, a few years ago I went through the following several times in a row:

  • Observe some curious combinatorial statement that did not seem to be well-known.
  • Later discover that it is somewhere in Richard Stanley's Enumerative Combinatorics.

So there were two options: either the trains of thought I had been pursuing had already been well studied, or Richard Stanley is a mind-reading time traveler.

Anyway, the only advice I can give about what to do in this situation is to become quite familiar with the basic results in the field. Then maybe talk to a trusted mathematician and ask whether the result sounds familiar or not. Perhaps pose the question you answer (without your answer) on math.SE and see if it's easy enough for someone to answer in their spare time.

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$%&* time-travelers! (+1) for, among other things, not shutting this guy down. –  The Chaz 2.0 Aug 5 '11 at 20:57
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At your age, there is no reason to worry about publications. In fact, it is extremely rare for people to publish anything worthwhile before graduate school. This includes top-rate people (Fields medalists, etc.). Publications play essentially no role in either college admissions or graduate school admissions.

One reason for this is that the areas of math which are accessible to high school students (or even undergraduates) have been mined extensively over the last 300 years. There are a few exceptions, but you won't discover them without extensive guidance from someone who is experienced. And even then, any papers which result will likely be of little importance. I know a number of people who wrote papers when they were undergraduates which they are now a little ashamed to include on their CV's!

This is a long-winded way of saying that without looking at it, I can can be 99.999% sure that your theorem is either 1. false, 2. trivial, or 3. known. But you shouldn't feel discouraged by this. Your goal in this part of your life is to learn as much math as you can. Certainly try your hand at proving things if you get some ideas. But it's not a good use of your time to worry about publishing things -- there will be plenty of time for that later. Just have fun and take the time to enjoy math!

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