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I am currently pursuing a PhD in an area of mathematical analysis. Recently I uploaded a result on arXiv where I somewhat improved a result from 40 years back in a question in smooth dynamics, a field different from my thesis subject.

In my limited interactions with experts in this field, demonstrating the existence of such a theorem with some parameters was considered important, and although the author clearly made an attempt to get the best possible parameter set for which the result holds, that in itself seemed to not be as important of a question.

A year back, by significantly extending this original author's methods, I improved the parameter set somewhat for which the result holds. This ended up taking a lot of effort, and I wondered while working on this, if I was only rediscovering what could already be known.

Recently, another paper appeared on the arXiv clearly improving my result significantly, proving what I ideally aimed to get but couldn't, building on some other papers that I was not aware of (probably leading back to an old result of Yakov Sinai).

While this is not in my thesis area and perhaps experts in my thesis area will not bother very much, does this still give any negative impression of my work in the academic job market (perhaps I should have done a more thorough literature search)? Also, does it give a negative impression of me for experts in the subject area of this paper?

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    An independent citation in mathematics often, although not always, worth more than yet another paper.
    – Matsmath
    Dec 22, 2023 at 2:31
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    If I understand the timeline correctly you improved a result that was 40 years old. And now this result was improved again after 1 year. I would see this as a great accomplishment. Someone read your paper, thought it was interesting and relevant, and sat down to further improve it. Dec 22, 2023 at 16:43
  • Actually, even though I completed the work a year back, I only put up the result on arXiv six months back, and the new preprint appeared just a few days back. Dec 23, 2023 at 22:43

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You advanced the field. That reflects positively upon you. The fact that someone advanced it beyond what you did reflects positively upon them. Credit is not a zero-sum game: Are the people who built the machinery that Andrew Wiles ended up using to prove Fermat's Last Theorem looking bad just because he, and not them, did the last step to prove the big result? Surely not -- they laid the stepping stones he walked on and now he, like Newton, can claim that if he was able to see further than others, it is only because he had been standing on the shoulders of giants.

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  • The issue is, the correct stepping stones were laid out, are known to a small set of experts, whereas I effectively ended up reinventing the ( not fully complete) wheel. But it could be my methods could be useful in other related questions, which would be interesting to figure out. Dec 19, 2023 at 23:03
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    @AritroPathak basically what I was going to say: whether your method was better than these other people’s or not, what you developed might still be useful to other people for other applications in the future. You never know who might adopt them. So long as it’s novel, you’ve still done valuable work Dec 20, 2023 at 0:25
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    "Credit is not a zero-sum game" 💯
    – Galen
    Dec 21, 2023 at 17:54
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Poincaré and Lorentz both came very close to discovering special relativity - so much so that some people claim they should get credit as co-discoverers of the theory. Similarly with Hilbert and general relativity.

Does that make these distinguished scientists look bad? Well, they missed out on the credit for discovering fundamental theories of nature. It’s certainly annoying to come close to an important discovery only to see someone else flesh it out in its full glory.

At the same time, scientific credit is, by and large, additive rather than subtractive: people care about what you did, not about what you didn’t do. That’s why Poincaré, Lorentz and Hilbert are still remembered for the many important discoveries that they did make (and even are seen as having made very important contributions to special and general relativity). Similarly, in your case, you can expect to receive fair credit for your improvement to the previously known result. The fact that it was subsequently improved upon by someone else may make you feel bad — trust me, it’s a feeling almost any researcher is familiar with — but it’s not something anyone is likely to hold against you in any way.

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    +1. Reading the first two paragraphs of the answer reminded me about a quote: "No scientific discovery is named after its original discoverer." I learned about the quote and many other similar remarks from the footnote #3 on page no. 98 from the book Algorithms by Prof. Jeff Erickson. Link. :) Dec 20, 2023 at 9:40
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    @medium-dimensional That is called Stigler's law (though of course not what this post is about)
    – Kimball
    Dec 20, 2023 at 13:59
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The fact that someone did better than you isn't a negative reflection on your work, actually. You may not get published if you don't currently have the best results unless your methodology is interesting enough on its own, independent of the new paper.

You might consider offering to collaborate with the other author(s) in further extending the work.

Much of math (all of math?) is insight driven. Perhaps you had (have) different insights than they do. Combining them can be synergistic and benefit everyone in the math community.

But I wouldn't take it as a negative. Even the best of us get passed eventually. That is a feature, not a bug.

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Couldn't you just update your arxiv preprint with a remark in the introduction that when you uploaded this originally, you were not aware of papers W and X, which were used in the meantime to prove Y in preprint Z? So that people who read it know that you are aware of this now?

I don't think chances that the original preprint will affect your reputation are high; still a remark like this may reduce the already low chances even more.

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  • Yes, I might make an update to the preprint, if and when it ever does get published, and make a note of this in this case. Making the point that perhaps the method which is very different, could have use in some other questions (some which I am trying to find). Dec 21, 2023 at 22:09

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