Is an easy proof of an interesting result worth publishing?

Question

I am a Phd math student, and I was wondering what was the general consensus on what is publishable and what is not.

I've come here with an example in mind which should give some insight. I recently proved an inequality which was needed for some other work, and it is a very crucial inequality. The proof generalises an already well known inequality, but in a quite easy way. I mean that the generalisation is both very easy to obtain and does not use any new machinery, it's obtained by just a modification of the proof of the original one. And I mean that if the authors put a remark below their proof saying "here do this and that instead of what we did, and the generalisation is easy", everyone would see how to do it.

Now, the proof is very short and basically follows the old one, but I thought it may be worth publishing because the result, although easy to obtain, is technically new and potentially of interest (again, it's not a surprising result, but still). But the proof is not interesting, it's basically a copy. I'll talk with the supervisor to see what to do, but I was curious about different takes more experienced mathematicians have on this. Would you put the result out there (obviously saying that you used the old proof almost unchanged) or leave it be?

EDIT: Apparently the generalisation is not known. I mean, people working on this are like "yeah of course in some way you can do it, 100%", but probably would need to at least think a bit on how to do it

Answer 1

The first answer to this sort of thing is to always talk with your adviser who will know more about the state of the subfield than you will, and will definitely know the specifics more than people here.

However, that said, my general inclination is to say that things like this should be published. First, it is often the case that something that seems obvious to you is not as obvious as one might think otherwise. Our own ideas often seem more more obvious since we have been thinking about them a lot. Second, having extensions even small extensions can be useful for other people have as actual references. Third, at the vast majority of points in one's career you will not be hurt by publishing a mostly straightforward result, and that's especially the case if one is still doing one's PhD. Worse case scenario, you put it on the arxiv and then after sending it to a journal they reject it as too obvious. (And to echo Andy's comment below do not put it on the arxiv without consulting with your adviser.)

That said, it is possible that the ideal place is not to publish it by itself, but possibly to publish it as part of some paper where one is proving larger results. But this again falls very much into a talk-to-your-adviser thing.

Answer 2

Is an easy proof of an interesting result worth publishing?

Probably not (but "probably" doesn't mean "certainly", so, as others said, ask your adviser).

The issue is that as a PhD student, your perspective on what counts as "interesting" and what counts as "easy" is likely to be different than the perspective of the sort of more experienced mathematicians who make up the intended readership of your prospective publication, and who would referee it. As one gains experience in mathematics, what one once regarded as "easy" will often get downgraded to "completely trivial" (and what one regarded as "somewhat nontrivial" gets downgraded to "easy", what one regarded as "hard" gets downgraded to "somewhat nontrivial", and so on). Similarly, what one has regarded as "interesting" as a PhD student will quite often later be revised to "mildly noteworthy" or even further to "not interesting".

Basically, this evolution in one's perception of how interesting/clever/publishable/etc one's ideas are is not only common and natural, but always points in the same direction. So if you think your result is easy now, just wait a few years...