Submit to an appropriate journal

Few, but ripe. (Carl Friedrich Gauss)

It is always tempting to submit a paper to a prestigious journal, but if the paper makes only a borderline case for publication in this journal, then the net result may be a lengthy process, critical reviews holding the paper to a very high standard, and ultimate rejection of the paper.

For instance, with JAMS, a paper really has to do something that makes referees excited and enthusiastic; a paper which is merely a good, solid application of mostly standard techniques to solve a moderately interesting problem will unfortunately have a rather low probability of being accepted into JAMS, even if it would have been readily published elsewhere. (Conversely, if the result is making people excited and enthusiastic, I do hope that you consider JAMS for your paper.:-) ) [UPDATE, 2024: the same statements would also apply to Forum of Mathematics, Pi, where I am now an editor, having completed my term at JAMS some years ago.]

Similarly, a journal devoted to research mathematics is unlikely to accept any paper whose primary focus lies in recreational mathematics, physics, philosophy, biology, computer science, or anything else outside the scope of research mathematics.

It is also a good idea to check that the editorial board of the journal you are submitting to contains at least one member who is expert enough in the fields that your paper is in that he or she can judge its quality appropriately and send it to a good referee. (But be cautioned that if a paper is too close to an editor’s interests – for instance, being heavily based on a paper of the editor – this may make it difficult for the editor to be objective and create some awkward conflict of interest.) Viewing a sample issue of that journal may also give you a sense as to whether it is a suitable venue for your article. You can also look at journals which have accepted papers similar to yours in the past, but of course there is no guarantee that they will do the same for your paper; indeed, if your paper is closely modeled on an existing paper in a journal, they may feel that the amount of new material in your submission may not be sufficient to warrant placing it on the same level as the earlier paper.

Generally speaking, it is not recommended to simultaneously submit two unrelated papers to the same journal; there is a possibility that they may somehow get confused with each other (for instance, a report for one paper may accidentally be applied to the other), and editors may not wish to give the impression of overly favouring one particular author in the journal. Also, if it ends up that the referee reports for one paper are more favorable than for the other (or if a referee makes a direct comparison between the two), it becomes quite likely that the paper with the less favorable reviews will be rejected. For two closely related papers, I would only recommend submitting to the same journal if it would make sense to have a single referee for both papers (but this can be quite a big request for a referee to accept).

If one is unsure what level of journal would be appropriate for your submission, consider scoring your paper as honestly as you can according to the following rubrics:

1. Correctness. Is the paper correct?
   1. Is the paper written so that it is easy to check that the arguments are correct?
      1. Are sufficient details given to check the arguments?
      2. Are all results and definitions specified accurately?
      3. Does the paper avoid relying on results that are not easily accessible in the literature, or for which the correctness is uncertain (e.g., unpublished preprints)?
      4. Does the paper have a modular structure that makes it easier to detect and correct errors, for instance by creating key lemmas, or by providing (either in full or partial detail) alternative arguments for some key steps?
   2. Are prior results in the literature cited accurately?
   3. Does the notation in the paper follow, or at least consistent with, those in prior literature, in order to avoid confusion? If not, is a justification of the change in notation given explicitly?
   4. Does the paper perform various “ sanity checks “, for instance by providing near-counterexamples to show that the results are sharp? Does the writing and structure of the paper demonstrate awareness of common pitfalls in the field (for instance by devoting particular detail to the subtle steps in the argument, and less detail to the routine steps).
   5. Has the paper been extensively proofread? Is it a polished final draft rather than a hasty first draft?
   6. (Secondary) Does the paper supply heuristics, numerical evidence, or analogies with prior results to convice the reader that the results are at least plausible, if not completely correct?
   7. (Secondary) Do the authors have a good track record of publishing correct results in this area (or at least issuing timely errata when issues arise)?
2. Novelty. Does the paper make substantial new contributions?
   1. Are the results new?
      1. Is there a broad array of new results (including minor variants of the main results that may for instance be included in remarks or discussion sections)? This should be weighed against the length of the paper; a very short paper that establishes just one result can be comparable in this metric to a lengthy paper that contains many results.
      2. Do the results form a significant advance over previous literature? Are they striking or surprising?
      3. Are there questions asked in previous literature that are answered in this paper?
      4. Are there many applications (or potential applications) of the new results? If so, are they addressed here, or in forthcoming work?
   2. Are the techniques new?
      1. Do the methods introduced allow for new proofs of previous results in the literature, or simplifications (or other improvements) of proofs of existing results?
      2. Would one expect the methods introduced to be useful for solving other problems?
   3. Is the narrative new?
      1. Does the paper add to the conventional wisdom, for instance by supporting existing beliefs, providing intuitive explanations for empirical phenomena that had been previously observed, or by introducing (or making more explicit) new principles and heuristics?
      2. Do the results go against the conventional wisdom? If so, are the reasons for this analyzed in the paper?
      3. Does the paper suggest new directions for research, for instance by posing new open problems and conjectures?
      4. Does the paper suggest new connections between existing results or fields?
      5. Does the paper clarify the strengths and limitations of various techniques, for instance by alowing for clearer comparison between existing methods?
      6. Is the introduction and other framing text written in one’s own voice?
3. Professionalism. Does the paper conform to professional standards?
   1. Is the paper written in grammatically correct and readable English (or another commonly used scientific language)?
   2. Does the paper have the standard format of a mathematics paper (abstract, introduction, acknowledgments, notation, main sections, (optionally) appendices and discussion, bibliography)?
   3. Are citations given whenever the paper refers to an existing result? If so, is the citation accurate and objective?
   4. Does the paper clearly distinguish between objective and rigorous arguments, and subjective opinions, conjecture, and speculation?
4. Presentation.
   1. Are the main results and contributions stated clearly and prominently?
   2. Is the paper structured in a logical fashion? Can the reader grasp the high-level nature of the arguments without having to read the paper line-by-line?
   3. Is the paper well-motivated?
   4. Is the paper accessible to a broad audience? Have efforts been made to lower the prerequisites necessary to read the paper?
   5. Does the paper focus tightly on attaining its goals (e.g., by avoiding devoting a lengthy amount of pages to unnecessarily digressions, or excessively convoluted arguments)?
   6. To what extent would a reader actually enjoy reading the paper?

Papers which score highly on a large number of these questions could be considered for a fairly top-tier journal; papers which only score well on a relatively small fraction of these questions might be suitable instead for a lower-tier journal. (But it can be worthwhile to “upscale” your paper by investing additional effort to improving its exposition in one or more of the above aspects, and then submitting to a higher tier journal than one might normally consider.)

The American Mathematical Society maintains a list of research journals in mathematics.

One note on rejection. There are a myriad reasons why a paper may end up being declined for acceptance by a journal; for instance, the journal may have multiple competing submissions and a target number of papers or pages to accept, and the referee reports may have been more enthusiastic about the suitability of other papers than the declined paper. In fact a positive rejection rate is to be expected if one is submitting to reputable journals; as I discuss here, I certainly have had my own share of rejections. I recommend not taking a rejection of a paper personally; use any feedback received to improve the exposition of the paper, and then resubmit to a different journal (usually moving from more selective to less selective journals in the process).