How do I know my results are worth publishing?

Question

I finished my bachelor degree in math and computer science (double major) recently and am applying for a PhD very soon. During this period I wanted to know how it feels to do research. Personally I am more interested in pure mathematics. As a result, I searched for around 20 open problems (not Riemannian hypothesis, small and unknown open problems) to see if I can achieve anything new about them to publish.

The problems did not need much to be understood, mostly in combinatorics, linear algebra and analysis. I was able to find new results about 2 of them, meaning that, I was not able to find any article that mentioned my results. However, I always feel my results are very bad and have low quality and I feel if I submit them to any journals I will only hurt myself. To explain a little bit more how my results look, I should say I was able to prove some special cases of the problem and find another elementary proof of one of the theorems that someone else found using advanced techniques. So by no means, I do not feel I did anything worthy of publishing.

However, when I am looking around the internet I see people publish results that are really easy to achieve, for example an elementary proof of Fermat equation for case 11. During my journey I found out achieving very good results will take years, those problems won't be solved easily and there is a reason that they are open for years if not decades.

So if we want to call a person a good pure mathematical researcher, does that mean solving very difficult conjectures only? It's really important for me because I am not sure if I want to go into the CS industry or do research on math and CS theory. Because I am afraid I am not a good fit for research and I feel that is because I have high standards.

Answer 1

As a novice, you probably don't (can't?) know if the results are worth publishing. Get some feedback from someone you trust, perhaps a professor.

No, mathematicians don't always work on or publish only the hardest of problems. Some of those have taken hundreds of years to resolve, not to mention new technology to assist. Some problems are very niche and contribute something of interest to only a few people. They can be significant, but the main interests of the moment are elsewhere. My dissertation was like that. One of it's main useful features is that it gave me insight into a class of problems.

Also note that you don't suddenly become an olympic level swimmer the first time in the pool. It takes years of practice and probably the aid of a good and observant coach.

If you think you have a worthy result you can take a small step and share it with some other more skilled mathematician asking for feedback. You can also take a large step and submit it to an appropriate journal and rely on the review process to get feedback on its structure and value. If it is terrible you will learn quickly in most cases. If it is worthy but needs improvement you will get some feedback on that (most cases).

But, you have to get in the pool to start the process, which you seem to be doing.

Answer 2

I'm not a mathematician, but I will try to explain you the general principle of deciding whether a research result is worth publishing.

Whether your research findings are a result of hard work or were derived in just an hour is pretty much irrelevant, and what's actually important is their significance. Ask yourself the following question: how will my findings impact the research area? How will they be used by other researchers? If you have something positive to say in response to these questions, then your result is worth publishing at least somewhere.

Next step: try to somehow quantify the significance of your research result in order to determine the best suitable journal. You can make a crude estimate based on the impact factor. Every journal has an impact factor, which is essentially how many times papers in that journal get cited per year on average within the two years following the publication year. So, ask yourself how many citations you expect per year. To figure out the answer, you can have a look at citation stats for papers that address problems of similar significance in your research area.

Also, keep in mind that different journals have different scopes and styles. There are journals for relatively short letters, and there are journals for long detailed articles.

One more thing to consider is that writing a paper takes time and effort, so you should weigh what's more beneficial for your career - spending time on writing up the results you have at hand or trying to get more significant results instead. And the choice here depends on your survival strategy in academia: some people publish like crazy to build a long publication list, while others strive to publish in top journals and achieve impressive citation stats. So, if you want to impress potential employers by a long publication list, publish everything you can - the more the better. But if you want to impress them by the quality of your research, then an almost uncited publication in a subpar journal won't help you much.

So, the issue boils down to what you are as a researcher, and no one is going to tell you exactly what you have got to be. You've got to decide about it yourself.

Answer 3

Something I will add is that its not necessarily your job to decide if something is publishable - or even interesting. That's the job of an editor and reviewers. If you have made some advancement then it is arguably worth you pursuing publication - small incremental publications are better than no publications in my opinion. It's hard to judge your own work, so submit it and leave it to the editor to decide