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I was asked by the editor of a mathematical journal to review a submission by an author (unknown to me) which heavily relies upon some of my earlier work. I do not have much experience with writing peer reviews, but I have a good grasp of the topic of the submission.

The new results in the submission build heavily on earlier results in my published work; they are interesting and, in my opinion, worth publishing. However, while reading the mathematical proofs, I couldn't shake the feeling that they were overly complicated. Indeed, after thinking about it some more, I found that the proofs can be dramatically shortened by using insights that the author may not have had. For instance, one proof would go from roughly three pages to around half a page. For another proof, it seems that it can be reduced to a more standard situation, again drastically shortening it.

My work-in-progress report on the paper now 1) sketches the shortened proof for the first situation, 2) describes the needed insight to reduce the proof in the second situation to the more standard situation. At this point, the suggested changes seem to amount to a significant contribution to the paper since they touch not just minor parts of the paper, but the bulk of it.

At this point, I wonder what the best way to proceed is. Should I just submit the report with these suggestions? Or may it be appropriate in this situation to suggest to the editor that, if the original author is interested, he may contact me so that we can co-author a revised paper together? Or would that be considered unethical?

I do not want to overstep my bounds as a reviewer, but on the other hand I don't know how common it is for reviewers to contribute significant improvements to a paper anonymously.

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  • Are you reviewing the paper anonymously?
    – Mindwin Remember Monica
    Nov 3, 2016 at 17:52
  • @Mindwin Yes, I do.
    – cfh
    Nov 3, 2016 at 21:17
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    An anecdote: one anonymous reviewer once rewrote a whole paper and said: is that the paper you intended to write? They didn't ask for coauthorship.
    – Captain Emacs
    Nov 4, 2016 at 1:01
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    @cfh then there might be ethical implications of breaking anonimity; conflict of interest. Once you are an author, you cannot be a reviewer by the eat/have cake lemma.
    – Mindwin Remember Monica
    Nov 4, 2016 at 11:40

4 Answers 4

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At this point, I wonder what the best way to proceed is. Should I just submit the report with these suggestions?

This is indeed the high road to take here. As a reviewer, you explain your ideas on how the paper can be improved. The authors will or will not take all or some of them into account.

Asking for co-authorship in return for your service would likely be perceived as very cocky and unprofessional. I doubt that either the editor or the authors would be pleased about this or seriously consider taking you up for your offer.

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    If the review is significant enough to warrant co-authorship (very rare - probably more substantial contribution required than what the OP describes) the editor will often intervene and contact the authors and explain the situation to the authors. It is extremely unheard of and inappropriate to ask to be a co-author yourself.
    – WetlabStudent
    Nov 3, 2016 at 15:05
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    +1 Offer of co-authorship is for the authors or possibly the editor to make, if at all. It is unwise for OP to ask, I am afraid. It is a bit of a shame that there is not a clear established code of conduct for very substantial improvements of a paper/proof which goes far beyond a regular review (and would, if done on the submitter's side, warrant co-authorship rather than acknowledgement) demanding that the reviewer be elevated to co-author, but that's the current state of things. Some journals/conferences publish prizes/commendations for reviewers.
    – Captain Emacs
    Nov 3, 2016 at 15:05
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    Note that I didn't consider "asking" for co-authorship, but merely "offering" to the author to contact me, which I feel does make a difference. Anyway, it seems your view has the majority behind it and seems to be the way to go. It irks me not to get credit for a contribution, but perhaps this is something one has to get used to when reviewing.
    – cfh
    Nov 3, 2016 at 15:14
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    @WetLabStudent Actually, the contribution described in the question seems easily significant enough that co-authorship would be a no-brainer in any situation except the contribution being made by a referee.
    – David Richerby
    Nov 3, 2016 at 22:44
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    @CaptainEmacs That's why I explicitly said "in any situation except the contribution being made by a referee."
    – David Richerby
    Nov 4, 2016 at 2:09
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I have been on the receiving end of such an extremely helpful review, and I am grateful for that. Since (s)he is anonymous all I could do was thank the anonymous reviewer at the usual "thank you footnote". That won't help her or him much, but if he or she reads it he or she will know that the help was appreciated.

However, the editor does know that you have written an exceptionally helpful review. Leaving good impressions like that does tend to come back at some point: maybe (s)he is in a hiring committee for a position you are applying for, maybe the journal is looking for new (associate) editors, ...

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    I would like to add that the author, for an exceptional review, as described by the OP, should probably say something more flattering than the standard line in the acknowledgments, something along the lines "We thank X anonymous reviewers for their comments, one of whom's significant insights greatly simplified the proofs of theorems X Y and Z". You don't often see such specific acknowledgments like that because most often the reviews aren't that significant. However, it is more common than you might think.
    – WetlabStudent
    Nov 3, 2016 at 15:00
  • Sure, what I did was add such a footnote at the point where (s)he made that contribution
    – Maarten Buis
    Nov 3, 2016 at 15:53
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    +1 for mentioning that accruing good karma will pay off one way or another.
    – Wolfgang Bangerth
    Nov 3, 2016 at 17:12
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    I once submitted a review and had the authors contact the editor to ask if I'd like to be a co-author of the revised version. The editor asked me, without revealing my identity to the authors until after I agreed. I also once completed a review (in my institution's internal review system) in which I suggested various improvements, one of which would require quite a lot of work which I was better placed to do than the author. Rather than doing the work myself, I said, "if you want a co-author to help with this, contact me." He did. I don't think I'd do this with a journal review, though.
    – Significance
    Nov 4, 2016 at 0:36
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In line with the other answers, I agree that you should not suggest becoming a coauthor.

However, an alternate route could be to relinquish your anonymity by signing your review. This would allow the authors, should they choose, to acknowledge you by name or contact you directly.

There is the possibility that your choice to do so could be regarded as self-serving in this case, but there is clear precedent for signing peer reviews.

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    According to your second link, 0.96% of peer reviews in mathematics are signed. Calling that a "clear precedent" seems to be a stretch.
    – Pete L. Clark
    Nov 4, 2016 at 20:09
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If the paper is otherwise sound and of acceptable quality, you could submit your review saying as much, and then also include your improved proofs as a follow-on submission to be published, ideally in the same journal issue. The original authors come out ahead, because not only does their paper get published, but also cited right away.

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    I think the results aren't big enough to warrant re-proving in a second publication.
    – cfh
    Nov 3, 2016 at 21:19

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