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A hypothetical situation: two (groups of) authors have independently proven the same result. How do we determine who gets credit?

There are some obvious bounds - if Group A posts a paper to the arxiv, and Group B posts the same result to the arxiv n time units later and claims to be independent; if n=10 years, then clearly Group A gets credit. If n=1 week then it's less clear to me.

Other than posting a paper to the arxiv, to what extent do talks count? What about talks containing nearly complete proofs?

ETA: I'm aware that such situations occur not infrequently; I only say "hypothetical" in the sense that I'm not personally involved in such a situation and I'm just curious to know the community's opinion.

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    What do you mean by "who gets credit"? If one cites this result both published within one week, then ideally they would cite both, if they are aware of both. – yupsi Nov 28 '17 at 19:47
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    Moreover, this is far from being hypothetical. I believe there was some problem in Group Theory where two groups published within hours in arxiv? – yupsi Nov 28 '17 at 19:48
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    Usually both get credits, and the result gets both names attached to it. – darij grinberg Nov 28 '17 at 19:51
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    It's not particularly strange to see papers doing the same thing posted almost simultaneously to the arXiv. Sometimes, researchers even coordinate this, once they realize that they are close to solving the same problem (the situation is usually less adversarial than it sounds, unless someone is trying to game it). – darij grinberg Nov 28 '17 at 19:52
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    @darij Or of course none of the names get attached to it and it keeps being referred to by the name of the original conjecture (though the KL conjecture might be unique in that regard). – Tobias Kildetoft Nov 28 '17 at 21:11
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This situation is by no means "hypothetical". If you are not alone to be interested in the range of problems you are working on, it is pretty much guaranteed to happen to you sooner or later.

The answer to the question as posted is "on a case by case basis". No general rule applies and no universal attitude exists. You can meet everything from bitter rivalry, fight for priority, and open hostility between the "groups" (or individuals) in question to a gentleman's argument of the kind "you go ahead and publish now and I'll hold my opus for a while".

The simplest and easiest to handle case is when the proofs run on drastically different ideas and employ nearly disjoint sets of techniques. Then one may, of course, try to squawk that he was there a few days earlier but everybody understands that the real test is whose approach turns out more viable and useful on the long run (and quite often they both survive, just lead to different developments). There is no question about the possibility to publish each of them either.

The worst case scenario is when the proofs are nearly identical or one is obviously superior to another, so once you see one paper, there is no reason to look at another one anymore. If the authors know each other and are on friendly terms, they can usually figure out some reasonable strategy (combining everything into one article under all names is the most common way out). Otherwise pretty much anything can happen.

Most situations are somewhere in between. So, just apply your common sense, remember that good relationships are more beneficial on the long run than establishing priority, don't think that you (or anybody else) may really own a mathematical statement or its proof any more than one can own the wind or the light or that one can carry any "credit" beyond the grave, enjoy other people successes as your own, and you'll be totally fine.

If the question is just about "how to cite properly?", when in doubt, cite both works and abstain from any judgement about priority, etc.

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Situation 1

Two Groups, Same result, Different procedure, Uploads or publishes at same time to arXiv.

Both of them get credit for their work. Especially the procedure that is easy to reproduce and more concise, understandable, intuitive and errorless gets more citations as years pass by.

Situation 2

Two Groups, Same result, Different procedure, One Uploads to arXiv before submitting to journal and other submits directly to a peer-reviewed journal, both of them gets published at same time in Journal.

Both of them get credit for their work. However, one who submitted to arXiv gets more outreach. Again here, the elegance of the work is still the predominant factor that determines who gets popular for the work.

Situation 3

Two Groups, Same result, same procedure, both uploads to arXiv.

Clearly trouble. Why? Plausible plagiarism. However, both of them can still get the credit if they mutually agree to collaborate. Other ways to resolve the conflict is to keep making progress in the article and write more and more on it beating the other group. This is a perfect way to resolve. As two groups who continue will now produce papers which will hopefully fall in situation 1 or 2. Or sometimes will produce Different result going in different directions.

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