Maximising the results-to-effort ratio

[Reprinted from a Google Buzz post of Feb 12, 2010; expanded, July 20, 2019.]

As a professional courtesy, research papers in mathematics should be at a “local maximum” with respect to the results-to-effort ratio: any “cheap” consequences, generalisations, variants, illustrative counterexamples, etc. of one’s main results should be put into the paper if this can be done with only moderate effort on the author’s part. If one is too lazy to do this, these consequences might not appear in the literature for some time (as they are too close to your own paper to be separately publishable in their own right), and each reader may have to rederive them by himself or herself, which is a much less efficient process in the long run.

Conversely, if a huge fraction of the paper is devoted to only a minor extension of the main results, one may consider removing that section, or replacing it by a sketch or even just a remark; it may be that a subsequent paper is able to achieve that result with much less effort anyway.

In a similar vein, if one is able to prove some new partial result towards an interesting open problem by a “cheap” argument, such a result can still be publishable if one has explored all the natural ways to try to improve the results using more “expensive” techniques and concluded that these do not give significant improvements to the “cheap” result relative to the additional effort one would have to expend. However, if it does seem that more expensive arguments would be able to give further improvements, it may make sense to hold off on publishing the cheaper result until it becomes clearer what the improvements are. In some cases, the improvement is so significant that it no longer makes sense to publish the cheap result separately, although if the expensive argument is based on similar ideas as the cheap one then it can sometimes make sense to still give the cheap argument at the start of one’s final paper to help the reader get acquainted with some of the ideas of the proof. In other cases, one could consider writing two papers, one containing the cheap result and one containing the expensive one, but with the former written in a way that motivates the latter, and possibly also with the former paper providing some useful lemmas or propositions that the latter can use directly.