What to do with a mathematical article rejected from a journal because the anonymous referee found a simpler proof?

Question

Some time ago, me together with a coauthor submitted a mathematical article to a journal. It was rejected because the anonymous referee found a simpler proof of our main result. He wrote his simpler proof (schematically) in his review. It is indeed simpler, but it is not obvious or trivial by no means.

Also, the simpler proof is only for the main result, which is the main, but not the only result in the article. Other, smaller results are not affected by the simpler proof at all.

Now I am thinking about what to do with this article. I still think that there is some value in our article as it is, even knowing that there is a simpler proof of the main result.

So, I am thinking about the possible options:

1. Throw away the article completely. Do nothing.

2. Submit the article to another journal. Include a disclaimer at the beginning or at the end of the article with the full explanation of the situation.

3. Publish the article on ArXiv without submitting it to any journals. Include a disclaimer at the beginning of the article with the full explanation of the situation.

In the case of options 2 and 3, there is also a question of whether to include in the article the simpler proof itself (not just to indicate its existence, but to give the full proof) attributing it to the anonymous referee. I am thinking not to include it.

Any suggestions about what to do with the article?

Answer 1

I would contact the editor and ask them to contact the reviewer to ask whether they would consider being co-author on a revised paper or at least be acknowledged for the simpler proof.

Answer 2

Based on your comment, let me suggest that you may have an opportunity to get your proof published, either by appealing to the editor or, with minor revisions, to another journal.

In mathematics, insight is, in many ways, the most important thing, not the actual theorem statements. If a proof offers insight into a wider class of problems (and you have the opportunity to say this) it might be more valuable in the long run than a "simpler" proof that offers less insight.

My own dissertation includes a proof provided by my advisor (and so credited). I had a different, more standard, proof, but his was very different and unexpected. The dissertation was published in TransAMS and the only questions I remember (this happened long ago) about the paper were about that proof. The overall structure of the dissertation was very interesting and made complete theory, but it was that one proof that excited people.

I don't know if the editor would be open to such an argument, some would not be, treating rejections as final, but if your assessment is correct then (IMO) the paper, including your proof, has merit.

In any case, a revision, stressing the additional insights provided by the proof should be worth a look at least and possibly worth publishing. But, you may have to spell it out pretty explicitly.

Answer 3

I’ve experienced something similar, but on the other end (I was the referee). The paper I reviewed proved a result that I would consider interesting, but the techniques are well-known and could be done by an undergrad summer research student. However, given that the journal wasn’t particularly strong I saw no reason to reject the paper. The author however insisted that one of his lemmas, which he took 10 pages to prove, was highly innovative and technically challenging. The fact is that the lemma is extremely well-known, with a standard proof that takes half a page. I included the proof in my report and told the author to just use it.

I have also experienced the same thing as an author, but my reaction was not nearly as negative as yours. I once wrote a 35 page paper and submitted to a very good journal. The paper was carefully read and the report was negative, but the referee thoughtfully pointed out that many of my computations can be significantly simplified. I followed their suggestions and it shortened the paper by almost 20 pages! It was eventually submitted elsewhere and accepted, and I am glad that the referee took the time to point out a cleaner proof to me.

It is easy to take things personally, especially when it comes to rejection of a paper that we worked hard on. However, try to appreciate the effort the referee put in to give you a better argument. I don’t think including the shorter proof will hurt anything except possibly your ego. The community at large will almost certainly appreciate a shorter proof over a long one.

Answer 4

This is not a reason to not continue to pursue publication further. Make good on all of the feedback you received, and send it to a new journal for publication consideration. There IS a right home for your paper. Do your due diligence and get it published elsewhere!