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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 $f(z) = e^z$.2011-08-05
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    If you're 18, then presumably you'll be starting college in the fall. In that case, show your math prof your work.2011-08-05
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    $e^{z}$ is not a counter example to the theorem if you consider the extended complex plane.2011-08-05
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    @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.2011-08-05
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    @Qiaochu Yuan: Yes, I forgot to state that $f$ must be non-constant. "Using the fact that a rational function defines a branched cover it should be straightforward to lift R to the desired path" Sorry, that went right over my head.2011-08-05
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    @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.2011-08-05
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    Just wanted to add that in the case of complex analysis, there are literally thousands of people who have done PhDs in the subject over the years, and they all had to start by mastering the fundamentals. So not only have these thousands of people had to think about the same issues, but they have developed numerous techniques over time to attack them. So the bar is going to be pretty high. Still it might be possible, with the help of a friendly mentor(s) and given enough to time and effort, to make a publishable contribution.. but it would be a major undertaking.2011-08-05
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    A lot of the comments here are quite negative regarding the possibility of proving new results for someone who is only learning the topic. I think it is important not to discourage the possibility of coming up with something that is publishable, even if it isn't groundbreaking. There are a lot of theorems around, but also many interesting questions that people might simply not have asked or studied yet. I managed to find and answer one in my third year at university, but only published the result many years later. The important thing is to talk to people in the field to find out if it's known.2011-10-21

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