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I got a rejection recently on a short paper that gave a fairly short proof that a conjecture of a well-known mathematician was false. The reviewer said that “while the results are new and interesting,” “the proofs are fine, but not very difficult, and the techniques are not new.” Nonetheless, the arguments previously evaded some top people in the field who were interested in the topic. My question is, to what kinds of mathematics journals should one submit significant results whose proofs involve some clever trick rather than a revolutionary method? Also, should one not strive to find the most transparent form of a mathematical argument?

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    Where was the conjecture made? If it was made in a journal paper, then perhaps you can write a "letter" ("correspondence," "short communication") to that journal containing your proof. – Joel Reyes Noche Apr 29 at 8:40
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    @JoelReyesNoche Good idea, but it was made in a book chapter. – mbsq Apr 29 at 9:24
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    To answer the very last question: yes you should always strive to find the most transparent way to state your proof. It will benefit everyone, will make you a better communicator in your field, and a better educator in your university. – user758469 Apr 29 at 10:52
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    There is a slight distinction between "conjecture of a well-known mathematician" and "well-known conjecture of a mathematician", and it may depend on the profile of the person who made the conjecture. That said: this is most probably "just bad luck of the draw" and perhaps one cannot extrapolate too much. – Yemon Choi Apr 29 at 16:51
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    @d_b: Such folklore results are indeed extremelly prevalent in many areas of mathematics (although we have no way to determine whether the situation described by the OP is one instance of this). One should probably add that these folklore results tend to be a huge pain in the ass for all people who are not experts in the field, but whose research has some non-trivial intersection with it. So the attitude not to publish something since it is "not deep enough" is probably not doing mathematics a great favour, in general. ;-) – Jochen Glueck Apr 29 at 19:33
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There's a good chance the rejection is because your results are not interesting enough for the journal. In other words, the journal is looking for not only new and interesting results, but also complicated proofs and/or new techniques.

So: submit to another, probably less prestigious journal. Alternatively, you can submit to a similarly-ranked journal and hope the peer review process results in acceptance, which is entirely possible, because this particular reject reason is a judgement call and different people will come to different recommendations.

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    I view the peer-review process essentially as a random Bernoulli trial, where the success probability p varies with the journal (and of course the paper). – Kimball Apr 29 at 16:43
  • @Kimball: Why "view"? It's true for ≈80% of papers submitted to good places. – user21820 May 1 at 11:31
  • @Kimball I know what you mean (that the outcome depends on the random reviewer), but wouldn't that also describe situations where the majority of a journal's reviewers are likely to love your paper? Take for instance a Bernoulli process with p = 0.99. I think we need to put some theoretical upper bounds on p as a function of journal and paper here. :) – Mateen Ulhaq May 1 at 11:32
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Keep submitting to other journals. If the result is indeed significant then you will eventually find a journal that will want to publish it. If on the other hand you continue to consistently get more rejections, that probably means the result is not as significant as you seem to think. Keep in mind that just because a well known mathematician conjectures something, that does not automatically make it interesting, and does not automatically mean that your counterexample “evaded top people in the field”.

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    But what do you do with "not as significant" results that have not "evaded top people in the field"? Isn't a "simple" theorem that takes a couple of days to solve also worth publishing? I've myself saved notable amount of time from fairly simple preprints on arxiv. How to identify the journals (if such exist) that provide peer vetting and a tiny bit of credit for such results? – Džuris Apr 29 at 20:35
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    @Džuris your result may or may not be worth publishing, I cannot say based on the information you provided. Certainly many papers are published thatcontain results that are not terribly significant but still somewhat interesting, and there are journals that are not as strict in their standards. So as I say, keep trying to publish the paper, and possibly think about whether you can improve it in various ways. You will get valuable feedback and hopefully a positive result eventually. – Dan Romik Apr 29 at 20:39
  • my question (which imo aligns with the OP question) was how to identify appropriate journals when you are quite convinced that your result is worth publishing but is nowhere near the top in significance. – Džuris May 2 at 13:58
  • @Džuris in the absence of specific information about reeearch areas etc, any low-ranking journal should be as good a choice as any other. mathoverflow.net/q/136261/78525 – Dan Romik May 2 at 16:48
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I would suggest that you focus your thoughts less on "the proofs are [...] not very difficult" and more of your thoughts on "the techniques are not new." What happened here is that the referee doesn't think your paper is interesting enough for the journal that you submitted to because it doesn't contain important new ideas. This is a perfectly valid reason to reject a paper from a strong journal.

The first thing to think about is whether you agree with the referee here, do you think that the paper does involve substantial new ideas? If you think there are new ideas, then rewrite the introduction so that it will be clear to a referee that there are new ideas and then resubmit to another journal of the same caliber. If you agree with the referee and think there aren't new ideas, then leave the paper alone and resubmit to a less competitive journal. The referee clearly thinks that your paper would be publishable somewhere, so if you just lower your aims a little you shouldn't have trouble publishing it.

Finally two random bits of advice. First, if the paper is quite short you might consider a journal that focuses on short papers (like Proceedings of the AMS). Second, make sure that your intro explicitly says "this is an open problem posed by X in year Y." If it's only been open for a year or two and it's solvable by a standard method, then that means it just wasn't as interesting a problem as the person who posed it thought it was. But if it's been open for 10 years, then that's evidence your application of these techniques is more clever than it looks at first glance.

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You could perhaps emphasize the importance of your proof, therefore making your paper more about the result than the proof itself.

Sometimes, some writing judo (turning the disadvantages into advantages) can also help. Instead of "the proof is simple" and "the techniques are not new", maybe the proof is "elegant in its simplicity" and uses "well-established techniques". Being up-front with it allows you to control the narrative. Of course, this type of reframing can only take you so far...

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This sounds to me like the reviewer provided a shoddy argumentation for their judgment. How very dare you make your proof easy to read, instead of obfuscating it into a complex unreadable mess that makes it look more impressive?

If I were you, I'd resubmit to a similar-caliber journal, and hope for a more fortuitous pull of the one-armed reviewer bandit. Alternatively, you could consider corresponding about this issue with the editor of the journal from which you got this review. Surely, they would agree that making proofs difficult shouldn't be a goal in itself, so if that were the only argument for rejection you'd have a good case for being reconsidered.

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    Sorry, but the first paragraph is nonsense. – Buffy Apr 29 at 11:09
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    Care to elaborate? I stand by that, given what I know right now of the review. It's possible that the full review gives a fuller picture, but rejecting a paper because the results are interesting, the proofs are correct, but the proofs are not difficult enough!? That is entirely unacceptable. – Wetenschaap Apr 29 at 13:30
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    Do you really think that mathematicians prefer "obfuscating it into a complex unreadable mess". I doubt that a reviewer would last long with such an attitude. – Buffy Apr 29 at 13:48
  • Hilarious! Wait, is this a real answer? – A rural reader Apr 29 at 16:28
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    Everything here looks to me like the referee rejected the paper for a quite typical and perfectly valid reason (namely, it's a routine application of known techniques and doesn't contain novel or interesting ideas) but made some poor word choices in explaining their reasoning. There's no reason whatsoever to think that the referee wants the argument to be obfuscated or that they'd be more likely to accept an obfuscated argument. – Noah Snyder Apr 29 at 18:48
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You could talk to the famous mathematician who made the conjecture. He could be more resourceful than you and he might know the proper journal for your contribution. Moreover, he seems to be the only one who can endorse your contribution.

Edit: if he is an editor, you can also submit the paper to him if the paper align with the journal's coverage.

He will be more than happy to read your paper and provide you feedbacks because your proof is smart and concise.

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    Or, more directly, submit to a journal where that mathematician is an editor. – Noah Snyder Apr 29 at 20:29
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    Even if it turns out that your results won't ever be published, the professor might fix the proof or change the conjecture in the next edition of his book, thanks to your hint. – usr1234567 Apr 30 at 7:13
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To complement the other answers: on one hand, yes, I have seen several instances where a proof (of a significant result) turned out to be "insufficiently impressive/interesting", and was rejected. On another hand, yes, this can mean that an interesting meta-fact was discovered, namely, that the result turns_out to be (construable as!!!) so easy that it's abruptly not-so-interesting. Yes, a bit ironic.

On a third hand, there is a possibility that some state-of-the-art methods, possibly not widely known, render a question "routine", even though to the perception of many it is not obviously answerable at all. To greatly exaggerate: not all products of large integers are "known/documented", but those of us hip to the grade-school algorithm (or better) will not be interested at all in seeing any particular product explained (without some further features of interest). So the "new" feature is there, but it's insufficient...

Unsurprisingly, in the end, there is considerable context-dependence/subjectivity in referees' appraisals of submissions. And, indeed, to some degree, editors care about whether a given paper will enhance, versus diminish, the "reputation" of the journal. "Impressive" papers are sometimes preferred... And, again, this is context-dependent and subjective.

So, yes, sometimes a (meta?) proof that an issue is, after all, (potentially) straightforward often does not lend itself to "publishability". :)

EDIT: I added the quotes to "publishability" because, after all, putting a document on the internet is a far more powerful form ofliteral publishing than anything achieved prior to 1985 or so. Yes, I know we're talking about various forms of status-certification and gate-keeping. Yes, some quality control also, but not only...

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  • "So, yes, sometimes a (meta?) proof that an issue is, after all, (potentially) straightforward often does not lend itself to "publishability" This sounds like the mathematical version of the publication bias that affects several scientific fields. In case that a question was previously thought to be interesting, the discovery that it is actually more straightforward than expected, is indeed useful and important and thus, almost by definition, publishable. [to be continued] – Jochen Glueck Apr 29 at 19:21
  • [continuation] I agree with you that, in practice though, such a "negative result" will often be regarded as less publishable - but I would like to object the subtext of the sentence I quoted, which seems to describe this state of affairs as perfectly reasonable rather than biased. – Jochen Glueck Apr 29 at 19:21
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    @JochenGlueck, I do agree that this state of affairs is not at all perfectly reasonable... but only meant to explain it in "human terms", to explain how it can happen, as opposed to, yes, why it probably should not happen. That kind of thing. Thanks for your clarifying remarks! :) – paul garrett Apr 29 at 19:46
  • Thanks for your response! It seems that I misinterpreted the answer a bit. :) Anyway, I think that your answer makes a good point about the underlying human motivation. (+1) – Jochen Glueck Apr 29 at 21:02
  • Say 20 years ago someone published a conjecture that seemed to require a complex and interesting proof. And for twenty years nobody figured out that there was actually a very simple proof that the conjecture was false. So there seems to be an argument here that the false conjecture can never be corrected! – gnasher729 Apr 30 at 15:49
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You might want to start by putting it on a preprint server.

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    Not technically an answer to OP’s question, but it’s actually great advice! – Andrea Apr 29 at 19:17
  • Which you can't really do unless you get someone else to vouch for you. – Džuris Apr 29 at 20:24
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    @Džuris What why? Everybody can upload to ArXiv..?! – usr1234567 Apr 30 at 7:11
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    @usr1234567 arxiv.org/help/endorsement – Džuris Apr 30 at 7:25
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    This answer seems to assume that the OP has not already done so, and it does not address the actual issue at hand which is about getting things published (which may be important for e.g. career reasons). Hence I do not think that in its current form it is very helpful – Yemon Choi Apr 30 at 13:42
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What you are discovering is that, in many cases, the proofs are more important than the theorems, especially when they bring insight or when there is a new technique in the proof that might be applied to other work.

It seems like your work is "a bit interesting" but doesn't have the potential impact to bother with publishing it. Especially when seen in competition with other papers.

For my doctoral research I worked on three problems. For one it was so easy to postulate and prove theorems (several per week) that it was abandoned as trivial. It was "cute" but, as they say, there was no "there" there.

You likely gained some insight in the work, but if the proofs were trivial, others before you likely did also, but didn't think it was worth publishing.

The comment of Joel Reyes Noche to transform it into a short note is a good one, as are the comments and answers of others to just try somewhere else are also possibilities.

But another possibility is to take what insight you got from this work and use it to extend to something with more potential impact.


The now current XKCD might apply here.

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    I would also say, sometimes a simple proof is valuable because it is simple. The idea is easier to grasp and apply to other things. And it's not always easy to come up with these things. – mbsq Apr 29 at 12:43
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    @mbsq : "I know for a fact that one other very famous mathematician gave a more complicated argument for a weaker result" --- is this fact prominently mentioned in the introduction to your paper? Also, have you considered submitting your work to the journal that this weaker result is published in? – babu_babu Apr 29 at 14:07
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    "but if the proofs were trivial, others before you likely did also, but didn't think it was worth publishing" I think I disagree with this reasoning. First, it is not consistent with the description given in the question. More importantly, though, I do not agree with the rationale "if a proof is simple, it's likely that others have found it before": in my experience, proofs have a very strong tendency to appear simple only after we know them (even more so if the person who came up with the proof took some care to write things down as clearly as possible). – Jochen Glueck Apr 29 at 14:28
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    @JochenGlueck I am often reminded of Pascal's apology for having written a long letter, because he did not have time to write a short one. – Gregory J. Puleo Apr 29 at 17:13
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    @mbsq I don't understand. If you've been in contact with B and he was willing to take the time to write you a two-page email, why don't you ask him whether your proof is interesting? – Elizabeth Henning Apr 29 at 22:18

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