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I have written a math paper and I think/hope it is ready for peer review. I am 40+ years removed from academia and have no access to 'publications' but I have learned from my many re-writes and I have tried to follow the format of a sample "Journal.tex" from the Journal of the American Mathematical Society for my abstract, subject classification, and bibliography.

I would like to have peer review of my work. How do I begin?

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Figure out what journal is appropriate. To do this, there are two aspects. One is essentially "readers interested in the articles published there are likely to also be interested in your work", the other similar level of how ground-breaking the results are. Without a subscription, you should still be able to see the abstracts of the recently published papers, and quite often you'll be able to access the content somehow (preprint on the arXiv, author's website, scihub,...).

Once you have identified the one journal you want to submit to, just look for their "instructions to authors" or similar, and follow them. This will typically be very straight-forward. They might ask you for your affiliation, but you can just respond with "Independent Researcher" there. Then patiently wait for the referee reports (expect something between 3 months and 1 year).

If you've chosen in an inappropriate journal, or if your draft appears to be very off (eg almost no references), the editor of the journal could forgo sending it out to review, and directly reject it ("desk reject"). So make sure that your draft is sufficiently polished, and that you've picked a reasonable journal. On the upside, a desk reject will typically happen much faster.

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  • My dream publication would be one that a famous author recommended to me many decades ago: The Journal of the American Mathematical Society. My paper introduces a new formula I developed for generating Pythagorean triples and then shows how it and other formulas can be used to find "triples on demand". My queries on Mathematics Stack Exchange seem to support the idea that my work is original, even after the 2300 years since Euclid. What journal might consider, without much laughter, the subject of A^2 + B^2 = C^2?
    – poetasis
    Commented Jul 12, 2020 at 14:28
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    The Journal of the AMS is one of the most prestigious math journals. It seems extremely unlikely to me that your results stand a chance there. Anyway, what specific journal is a good choice for your manusscript is off-topic here.
    – Arno
    Commented Jul 12, 2020 at 14:57
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    My advice would be to share the paper with a professional mathematician for feedback before sending it off. Stack exchange would be only the first step to ask around whether a result is novel and important. Because there are so many crackpot amateur submissions it will be a challenge to get a professor to look at it even for 5min... But it should be doable if done right! Do you have a local university with a math department? Commented Jul 13, 2020 at 10:39
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I suggest the following tasks for you:

Dear Prof. X

I got this result and I posted this preprint (URL and printout) and I noticed that you recently published Y and I would be very greatful to you for any comments or some advice on where to publish it.

You will probably hear back from some but not all of the people you contact.

  • if you can turn it into a good contributed paper or a poster at a joint AMS /MAA meeting or a section meeting, consider submitting your proposal and attending if it accepted. I'm not sure how soon they'll have these again because of the pandemic. There's a chance you'll talk to people interested in your work and hear some feedback.

  • figure out a reputable journal that doesn't mind if authors upload preprints. (You will probably receive solicitations from not so reputable journals that will publish almost anything as long as the author pays; they won't be able to get you a good peer review.) It is very likely that you will get fair and diligent reviews, even if your paper is never accepted.

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    Registering on all of those platforms can certainly be skipped. But you are right that uploading a pre-print and sharing it with experts in the field is certainly the way to go.
    – Clément
    Commented Jul 12, 2020 at 2:44
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    If you are not affiliated with a university, you'll need to be endorsed before you can submit to the arXiv. So submitting a preprint before sharing the draft with anyone is not going to work.
    – Arno
    Commented Jul 12, 2020 at 11:34
  • @Arno I'm not familiar with endorsement or arXiv. Perhaps this can be taken "offline", meaning, if you send an email to [email protected], I can reply with my current PDF and, glancing at it you can suggest further action on my part.
    – poetasis
    Commented Jul 12, 2020 at 14:21
  • @poetasis A good discussion of arXiv ebdorsement is here. Commented Jul 12, 2020 at 16:50
  • @poetasis You seem to have already posted your result here. Not assessing your result, you should be concerned about someone trying to steal it. If arXiv is too challenging, you should try other preprint sites such as OSF. Commented Jul 12, 2020 at 17:00

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