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Would doing so go against the declaration many journals require that the author has not submitted the work anywhere else?

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It depends on the journal. If it is not found on their web site, ask them.

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    I'll note that in some fields the journals tend to be consistent one way or another. Math journals tend to be permissive. But, yes, ask to avoid problems.
    – Buffy
    Commented Jan 17, 2021 at 22:45
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You have to carefully read to be sure what "submitted anywhere else" really means. Depending on the field (e.g. math or physics), this might only mean that you have not submitted to any other journal, while posting it to a preprint server such as the arxiv is fine. For example, Physical Review Letters states

The manuscript has not been published and is not being and will not be considered by another journal while it is considered here.

clearly stating that this only refers to journals, and Nature Physics states

Nature Physics supports the posting of submitted manuscripts on community preprint servers such as arXiv and bioRxiv. We do, however, ask you to respect the following summary of our preprint policy:

  • The original submitted version may be posted at any time.
  • The accepted version may be posted 6 months after publication.
  • The published version—copyedited and in Nature Physics journal format—may not be posted on a preprint server or other websites.
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    BTW, interesting find while searching for the Physical Review policy on the arxiv: aps.org/publications/apsnews/201211/preprint.cfm. Posting preprints can come with unexpected risks.
    – user151413
    Commented Jan 17, 2021 at 23:37
  • Nature Physics ought to update the last bullet point in their preprint policy. They just started allowing authors to publish under a CC-BY license, in which case anyone (presumably including the authors) should be able to freely share the published version.
    – Anyon
    Commented Jan 18, 2021 at 0:13
  • @Anyon Though this probably means that you have to pay publication charges? The same holds for the hybrid and open access Phys Rev journals, I believe.
    – user151413
    Commented Jan 18, 2021 at 0:38
  • Yeah, and it'll cost a small fortune (unless using their Guided OA system where it's considered for three journals at the same time). I just think they should qualify that statement.
    – Anyon
    Commented Jan 18, 2021 at 0:44
  • @Anyon Oh, indeed, I remember seeing insanely high figures.
    – user151413
    Commented Jan 18, 2021 at 0:44
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Usually, yes, you can.

arXiv is a preprint server. Preprints are not like submitting a paper for review at another journal, they do not involve formal peer review, and arXiv is not a journal. Many (most?) publishers have explicit policies that deal with preprints as well, and they are usually minimally restrictive, e.g. Elsevier:

Preprint

  • Authors can share their preprint anywhere at any time.
  • If accepted for publication, we encourage authors to link from the preprint to their formal publication via its Digital Object Identifier (DOI). Millions of researchers have access to the formal publications on ScienceDirect, and so links will help your users to find, access, cite, and use the best available version.
  • Authors can update their preprints on arXiv or RePEc with their accepted manuscript .

with caveats

  • Some society-owned titles and journals that operate double-blind peer review have different preprint policies. Please check the journals Guide for Authors for further information.
  • Preprints should not be added to or enhanced in any way in order to appear more like, or to substitute for, the final versions of articles.

So you are usually free to post an article on arXiv, even before submitting it for peer review. Exceptions can apply if the journal you submitted to operates double-blind peer review.

Your question asks about something else, which is whether journal policies that say "has not submitted the work anywhere else" prohibits you from submitting to arXiv. Usually it doesn't, because this journal policy is typically of the form "if you submit here, your paper is not currently under review by another journal". Since arXiv is not a journal, there is no issue. Again here's another policy page from one of Elsevier's journals:

Submission declaration and verification

Submission of an article implies that the work described has not been published previously (except in the form of an abstract, a published lecture or academic thesis, see 'Multiple, redundant or concurrent publication' for more information), that it is not under consideration for publication elsewhere, that its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out, and that, if accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder. To verify originality, your article may be checked by the originality detection service Crossref Similarity Check.

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