I’ve never written a research paper before, but here are some thoughts:
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You can’t just upload your paper to arXiv: you need to be endorsed by someone. This bit of gatekeeping is to make sure people don’t just spam arXiv with crap. See also https://arxiv.org/help/endorsement
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I would just start writing the paper and see where you end up. Becoming proficient at writing is mostly a matter of writing a lot. The first draft of your very first paper is probably going to suck. It’s part of the process.
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I think it’s a good idea to share your first draft here on the forums to solicit feedback. You may find someone who is willing to improve the paper / co-author it with you. (But do provide some kind of first draft already. It’s much easier to find collaborators if you’ve already made a start.)
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You can find the typical format of a paper online. Typically it’s something like this: You start with a description of the problem and a short summary of your solution. Usually there is a section on related work. Then you explain your method in detail. Next, you show the results of experiments. Make sure to also include experiments that show it’s really your new method that creates the improvements and not some random thing (ablation study). Summary and conclusion. References. Appendix or supplementary material (if any).
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Use LaTeX or Word (but preferably LaTeX).
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It might be a good idea to start with a blog post rather than a paper. You need less rigor for a blog post and it will help to find a co-author.
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What is needed in the actual contents of the paper (i.e. math) depends on where you’re going to submit the paper (like a particular conference or journal). If there’s not enough math, the reviewers for that conference / journal will probably tell you.
Anyway that’s all I have for now. 