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I'm new to research. I'm working on a certain problem that has sort of been abandoned by all researchers (as far as publications go). Anyway, I like it for its simplicity and easiness to grasp. However, it's a new algorithm for a problem and a new way of possibly answering theoretical questions about the problem.

There is so little research out there on the certain problem, that it would literally take you a day or less to view it all and ascertain that indeed the idea hasn't been looked at yet. I'm applying the well-studied field of integer programming / linear algebra to model each instance of the problem.

Anyhow, regardless of whether I answer a major open problem, I think the approach is valuable in itself, if for no other reason that it makes answering basic questions about the problem a lot more easy, sort of like how algebraic geometry answers questions about geometric objects using algebra.

How do I proceed? Do I just spend a month perfecting the paper on my own and put it on arXiv? Is that all?

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    You need endorsements to put papers on arXiv. If you have those, then sure you can do that. You shouldn't do (just) that. arXiv is not a journal. You don't need to do anything special to submit to a journal. You don't need to be affiliated to a university. You can just try to find a relevant journal (e.g. the ones that contain the research you read) and make a submission. There should be a site for each that describes when and how to do it.
    – Derek Elkins
    Aug 2, 2019 at 22:27
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    There is so little research out there on the certain problem, that it would literally take you a day or less to view it all and ascertain that indeed the idea hasn't been looked at yet. --- Perhaps a statement like this can confidently be made in your field, but in mathematics even if it were the case that you could look over the research on something in a day, it could still take months or even years to shift through the literature (MR reviews, Zbl reviews, JFM reviews, things like this that few people seem to know about, etc.) Aug 3, 2019 at 16:31
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    to find those handful of papers, books, unpublished but circulated notes, etc. and to be reasonably sure that you haven't missed something that has since been mostly forgotten (buried in someone's Ph.D. dissertation or something). Aug 3, 2019 at 16:33
  • Thanks all for your guidance. I am actually working with someone who's persuing a master's in math. So, there's my "accademic connection." Perhaps he could show the paper to his professors for endorsement. Aug 3, 2019 at 20:01

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You are on the cusp between math and CS, actually. The better way to "publish" your work is to write it up formally, with references/citations and everything and submit it to either a journal or a conference. Choose one that matches the content of your paper.

But also make sure it is a reputable journal and not one that just publishes anything for the money you pay them.

However, you also have to be prepared, with journals, for publication fees, even with reputable ones. Sometimes they may waive the fees, but you may need to find a way to pay them. And, for conferences, make sure you have some way of attending if your paper is accepted for presentation. In CS, conferences are usually preferred. In math, usually journals.

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    However, you also have to be prepared, with journals, for publication fees, even with reputable ones — Really? I have never paid a fee to publish a journal paper. (I work at the intersection of math and CS.)
    – JeffE
    Aug 4, 2019 at 17:04
  • @JeffE, yes, a number of high quality journals charge page fees for publications. Often grants pay them. Often universities pay them. Sometimes authors do. Sometimes they are waived. I'm pretty sure TransAMS does (or did, back when).
    – Buffy
    Aug 4, 2019 at 17:07
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    Strictly speaking, Transactions doesn't charge page fees for publication; they charge a per-article fee for open access, which I find rather silly given the existence of the arXiv. (When I first started published 25 years ago, it was common for CS journals (SIAM J Computing, for example) to advertise voluntary page charges. I took the word "voluntary" at face value and never paid them. Nobody I knew ever paid them. Our papers were published anyway.)
    – JeffE
    Aug 5, 2019 at 4:10

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