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Assume I just wrote a paper. Now I would like to publish it in a journal.

What kind of things I should do before its submission? Should I first give a few talks at seminars or conferences? Upload to arxiv? Send to a few experts?

What is the best algorithm?

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    $\begingroup$ here is an overview of the process --- main message: Don’t try anything without advice and guidance from your advisor or an experienced mathematician familiar with your work. $\endgroup$
    – Carlo Beenakker
    Commented Dec 28, 2022 at 9:41
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    $\begingroup$ Perhaps academia.SE would be a better fit? There are a number of experienced mathematicians active there. $\endgroup$
    – David Roberts ♦
    Commented Dec 28, 2022 at 10:05
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    $\begingroup$ Go back and check through it a second and then a third time for typos. Too many experienced mathematicians skip these steps. $\endgroup$
    – Christopher Drupieski
    Commented Dec 28, 2022 at 13:53
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    $\begingroup$ Another general advice, hence only a comment: After you think you have completed your paper, take a break from it / do something else for a week or two, or three, then get back to it to reread it with a fresher/outside perspective. You will be surprised how many things you are going to notice and will be wanting to fix/edit/improve. $\endgroup$
    – M.G.
    Commented Dec 28, 2022 at 14:46
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    $\begingroup$ @DavidRoberts I looked at academia.SE several times in the past, and I found the amount of pointless comments and answers (pointless because the site is not maths-specific and a lot of people feel that they should offer their opinion even if it is not relevant) very frustrating. So I totally understand at least one reason to post this question on MO. $\endgroup$
    – Vladimir Dotsenko
    Commented Dec 28, 2022 at 14:59

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First of all, upload to the arxiv. Then you can write to the experts you know, and inform them that you uploaded your new paper. Then wait a little to give them opportunity to send you comments. If there is a possibility, give seminar/conference talks, but it is not necessary to do this before you send it to a journal.

Edit. Posting the paper in the arxiv is really the most important step. See the comment of Timothy Chow for many reasons of this. From the point of view of dissemination of the result, it is more important than publishing it in a journal. Some people who do not care about grants and promotions even do not bother to send their results to journals.

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    $\begingroup$ Just to elaborate a bit on the first point: Uploading to the arXiv does a few things. It establishes priority, in case someone else comes up with the same ideas almost simultaneously. If you give a talk, anyone who is interested in your work will want to see your paper, and it is convenient if the paper is already on the arXiv. Lots of people watch the arXiv, so you may get some very quick feedback from people who you might not have expected to have any interest in your paper; this is almost always a good thing. $\endgroup$
    – Timothy Chow
    Commented Dec 29, 2022 at 0:10
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    $\begingroup$ In my experience, posting on the arXiv gives you very little feedback. You are much more likely to get helpful feedback by emailing a draft to a few friends and experts in the field. On many occassions, I've also gotten helpful feedback from referee reports, in some cases saving me from embarrassing (fixable) mistakes; posting on the arXiv cannot be undone. Further, contrary to the idea that it establishes priority, I've actually seen a few results "scooped", because they were available on the arXiv and someone found stronger results and published those extensions quickly. Also... $\endgroup$
    – Pace Nielsen
    Commented Dec 29, 2022 at 1:18
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    $\begingroup$ the papers on the arXiv are not always trustworthy, not having gone through any refereeing. Many mathematicians do not trust papers on the arXiv to the same level as papers in good (reputable) journals. $\endgroup$
    – Pace Nielsen
    Commented Dec 29, 2022 at 1:20
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    $\begingroup$ @AlexandreEremenko My experience with double blind refereeing is similar to yours, but in the last year or two, I've twice been asked to referee a paper with the names of the author(s) stripped. It seems to be on the rise. Regarding Pace Nielsen's comments, yes, sometimes that kind of "scooping" happens, but I've also been in a situation where if my coauthor and I had waited a month, we would not have been the first past the post. I think that for most people it's better to post to the arXiv sooner rather than later. If someone quickly builds on my work, I consider that a good thing. $\endgroup$
    – Timothy Chow
    Commented Dec 29, 2022 at 1:26
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    $\begingroup$ @AlexandreEremenko If you are arguing that one paper has more value than another, based merely on the fact that one was submitted to the arXiv a few seconds before the other, then I strongly disagree. If you are arguing that this establishes priority, I also strongly disagree. (Given how long it takes to write a paper, it is obvious that these results are independent of one another.) Moreover, I'd argue that not only in math, but in all science, we should work to get away from such posturing. $\endgroup$
    – Pace Nielsen
    Commented Dec 29, 2022 at 22:09
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First of all, proofread it several times. Remember that every minor mistake that will take you five minutes to correct may perplex the reader for several hours, so aim at minimizing the community time, not just your own time.

Second, check that all references that you use in an essential way are easily accessible and contain full proofs, not just further references to endless "trees of knowledge". Remember that the full length of your paper is the sum of the pages in the main text, the texts you refer to, the texts the texts you refer to refer to, and so on until you reach some "common knowledge" layer (what exactly constitutes it depends on whom you are writing for, but don't aim just at "experts").

Third, show it to some knowledgeable people you trust and discuss with them both the content and the readability. You may get some useful feedback on both accounts.

Then, if everything looks good, place it on arXiv and send the link to the people who you believe may be interested. Meanwhile think of where you would like submit it. Once you have made your choice, submit following the chosen journal procedure (you can usually find all the requirements, style files, etc. on the journal webpage).

Once you get a referee report, see what you agree with and what you disagree with. Remember 2 things. First, the referees see your paper with the "reader eyes" as opposed to "author eyes" you see it with, so if they stumble somewhere and say that they would appreciate a clarification, then, most likely, some other readers will appreciate it too even if it looks crystal clear to you. Second, the referees are not all-powerful and omniscient gods, so it is totally OK to disagree with them both about mathematics and about stylistic issues. If you think that some referee suggestion is totally outlandish or stems from complete misunderstanding of what was written, you are by no means obliged to follow it. Just explain in the letter to the editor why you think so.

Once you get the galley proofs, go over them and compare with the submitted text. Check that all technical editor corrections make sense (usually they are minor, but even a minor change can alter the meaning of a sentence sometimes). If you are happy with everything, that's the end: the paper is out for publication and you do not need to do anything else. If not, report the needed corrections to the editor. Usually they just accept them at that stage.

That's the full process from the beginning to the end as I see it.

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    $\begingroup$ I'd only add that for step #4 (putting it on the arXiv), you would also want to follow the advice from step #3 and first talk to knowledgeable people you trust. There are sometimes reasons not to post a paper to the arXiv immediately; such as if you are still working on extensions of the result for a second paper that isn't yet finished (and you wouldn't want that work scooped), etc... $\endgroup$
    – Pace Nielsen
    Commented Dec 29, 2022 at 22:21

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