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?