Spot a possible improvement when reviewing a paper In reviewing a paper I notice that the result presented there can be improved. However, the improvement may not be significant enough to appear in a good journal. Therefore, to me the best situation would be to have the authors' permission and to join the paper under review.
I would like to be advised of what I may do. I think I may write to the editor, and I would especially like to know how to ask politely. Would anyone have similar experience?
Edit: First of all thanks a lot for all suggestions! I just forgot to mention that my report was already sent one month ago. Would there be something extra I should take care of?
Edit 2: Again, many thanks to you guys! Actually contacting the authors is never an option in my mind. My options are only: (i) tell the editor (not recommending myself as a coauthor); (ii) write my own note.
Here note means a short paper, since my new short proof may be of ten lines (the original proof is three pages), that's the reason I would like to consult you guys, and to see if there is a third option.
At the moment, I prefer to do the first option, and I am thinking of whether in the email to editor I should include my new proof, or just say &quot;I may further provide the details&quot;. How do you guys think?
Contacting the authors directly is a terrible idea. Don't go there. Contacting the editor with a question or request is fine. If you contact the authors directly and they interpret it as a form of poaching and complain to the editor, you may lose the opportunity to review in the future.
But the &quot;best&quot; option isn't to join the paper. It is to give your best review, pointing out what you think has been missed and asking for major revisions if that is what it would take.
Tell the editor that you would be willing to work with them on the improvement with permission and await what happens. You may not get a co-authorship (I doubt you will) and your advice might even be attributed to the anonymous reviewer, but that is pretty natural.
You will have contributed to the advancement of the art.