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Our team finished a paper 3 months ago. However, we use a result from another paper in a very top journal (one of the best). There are quite a lot misprints and somewhat mistakes. We can fix most of them but one inequality which they didn't have explanation why this inequality is true. We have spent 3 months to ask people (some extremely top mathematicians, they published many papers in the top 5 journals) and no one understand...We also asked the authors, but it seems that they don't want to explain that to us...We are really lost and don't know what to do...Shall we submit our paper? We really can not make sure that the inequality is true...

PS. This result is crucial in our proof...

Now, we have clarified the 'gap', there are some problems in the proof, the assertions are not accurate/correct. However, luckily, the final result is correct. After discussing with one of the authors, we confirmed that it is a gap and it took us quite some time to fix it.

Well, some people told me that the 'big guys' do not care about the details. I am not sure whether these incorrect details would lead to wrong theorem. As a young scholar, I have to say that I am a little bit disappointed at math research (well, maybe my field).

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    You use results from another paper that you do not fully understand in your own paper, is that correct?
    – BPND
    Commented May 29, 2017 at 7:41
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    Is there any proof of the inequality? Also, have you looked at any followup papers either by the authors or other people? Sometimes, later work explains some ideas better than the original authors...
    – Nick S
    Commented May 29, 2017 at 9:56
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    I wouldn't use the inequality. I think it's okay to use a theorem you don't fully understand (I've had a colleague who used a theorem with a prize-winning proof that took a long time for experts to verify). Yet, there are so many red flags here that go beyond you merely using an established theorem that you don't understand to the depth that the original authors do, you risk publishing an error. I think your best non-mutually exclusive options are: wait and see if the proof is published, or try and prove/disprove the theorem (in the second case, publishing the result independently).
    – Dr. Thomas C. King
    Commented May 29, 2017 at 11:04
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    Explain the inequality to a few physicists. If they believe it while they wave their hands around, and murmur something about Maxwell equations, you can use it.
    – Davidmh
    Commented May 29, 2017 at 21:27
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    If I were in your situation, I would explain, early in the paper (probably in the introduction, not in the middle of some proof) that I need this inequality, and I would give the inequality a name or equation-number. I would add that the inequality is asserted in [reference] but without proof, and I have been unable to find a proof. Then, in any theorems that depend on the inequality, I would explicitly assume it, by name or equation-number in the statement of the theorem. It doesn't seem pleasant, but it's honest.
    – Andreas Blass
    Commented Nov 4, 2017 at 0:21

4 Answers 4

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Should you submit your paper under this situation - possibly.

People assumed that Fermat's last theorem was true for centuries before it was proved. You don't necessarily need every detail of every proof.

If it seems reasonable that it is true then it's okay so long as you are clear on this assumption, and caveats thereof, and your paper is essentially going to be retracted if the inequality isn't true, or have conditions to it.

From your description however There are quite a lot misprints and somewhat mistakes this tells me that the paper is of questionable if not poor quality, and your mistake was doing further work that depended on this. In light of this I would not wish to publish without at least some verification of the inequality. Peer review in no way means something is true.

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    Agree. Peer review doesn't mean it is true. Especially for some `big' guy's papers....Most journals and reviewers would not question much about these papers.
    – user92646
    Commented May 30, 2017 at 7:00
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Ask on mathoverflow!

It's a site like this, but for math at research level. There is incredible expertise in all fields of math (I know of) over there. Before you ask, look at "How to ask" and also take a look around at the site to get a feel for how the folks roll.

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Well isn't it obvious?

You just say "This is a proof dependent on the truth of result X".

That will be true no matter if X is true or not.

And then it is still a useful paper.

Then if X is shown to be true, then your paper becomes a true proof.

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    It is not obvious. As word spreads that the results in paper A are not reliable, responsible researchers stop citing it, and other papers building on it end up being ignored as well. The end result is that, even if statements in paper B are clearly labeled as being dependent on the unverified parts of paper A, the usefulness of paper B may become negligible. Even if later the results of A are verified, people in the field may still remain ignorant of the results of paper B.
    – Andrés E. Caicedo
    Commented Oct 1, 2019 at 20:55
  • Well if that's the case, the paper will be rejected and no harm done. Anyway, result X could still be proven by someone else and then your paper is then a complete proof. Unless the person is concerned with being "the champion who put the final piece of the puzzle in the proof".
    – zooby
    Commented Oct 1, 2019 at 21:26
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If the paper clearly states the inequality is true and it was peer-reviewed and published in a reputable journal, you can assume it is true. The rule of thumb here is that you need to be able to fully understand the results of the paper you use, but not necessarily be able to reproduce ever result you cite.
For example: when in any field you use statistics (e.g. an inequality), you need to understand the final equations you are using. However it would be silly to require almost every scientist to be able to independently prove each and every equation from scratch.
The same is true for various measuring machines. As a scientist you need to know how to use your equipment and what it can and can't do. However beyond that, it is perfectly acceptable to cite that it works and treat it as a black box.

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    But apparently the OP is unable to "fully understand the results" which is exactly why he asks how he should handle it. I slightly disagree with your first sentence. You shouldnt "assume it is true" if it is not proven.
    – BPND
    Commented May 29, 2017 at 9:05
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    I disagree with this answer. This might be good advice in some fields, but for mathematics, this advice applied to this situation is in my opinion a big NONO. The OP wants to use an inequality which, from what I understand in his post, was not proven in that paper and was not referenced to a reliable source. And that raises a big red flag: in mathematics a result like this should be published only when it is obvious, which seems it is not. The fact that the authors refuse to explain it is another red flag... If you combine this with "here are quite a lot misprints and somewhat mistakes"...
    – Nick S
    Commented May 29, 2017 at 9:44
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    , which means that the authors made many mistakes on simple things (so why should one assume that they didn't make a mistake on this more complicated thing?). Given all those red flags, the paper should probably not have been published (at least not in this form) and unless they find a more reliable source I would be hesitant to rely on this.
    – Nick S
    Commented May 29, 2017 at 9:49
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    If the paper clearly states the inequality is true and it was peer-reviewed and published in a reputable journal, you can assume it is true. — This is absolutely wrong. At most you can assume that a few people think it's probably true, modulo a bunch of picky details that they didn't bother checking but smell okay.
    – JeffE
    Commented May 29, 2017 at 19:27
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    However it would be silly to require almost every scientist to be able to independently prove each and every equation from scratch. — True. But reading, understanding, and verifying a proof is a lot easier than proving something from scratch.
    – JeffE
    Commented May 29, 2017 at 19:28

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