BOOK CHAPTERS / PROCEEDINGS

HINTS ON WRITING A (SCIENTIFIC) REVIEW (OF A JOURNAL PAPER)

Particularly younger colleagues might wonder how to write referee report for a journal paper under consideration for publication in some scientific journal. There is no golden rule, however, some people might appreciate the following list of guidelines that I follow roughly.

The following points need to be clarified

(1) Is what is written there correct?  (2) Does it already exist? (3) Is it interesting? (4) Is the work sufficiently well written? (5) Conclusion

(1) Is it correct?

As a rule of thumb, what is quoted from elsewhere (if not obviously questionable) is correct. You should make sure that the quote comes from a source that has either already been published or accepted. Substantial things quoted from the arXiv are not good and problematic. If it's too massive, you have to ask the authors to elaborate on it. Are the arguments well written? Are the arguments understandable? You can safely see yourself as the target audience, i.e. you should be able to understand it (to a certain extent, this is the responsibility of the editor (for instance, me) to trust you with the task). If that seems impossible to you, you have to ask questions or point out mistakes. You are not there to correct mistakes. However, questionable conclusions should be highlighted.

(2) Does it already exist?

This is always relatively tricky, especially if the authors only cite a few things. I have often had cases where introductions and complete proofs have been copied from other sources. This is often hidden under phrases like 'as in [xx]', 'we follow the arguments in [xx]', `the following is inspired by [xx]'. Of course, you don't have all the literature in your head, but you might have an intuition that, although it seems obvious, it is relatively clear that someone has already done it. The impression that it already exists can arise in (at least) two different ways: First, the extension or generalization (if it is one) is very obvious for the current state of mathematical technology. For example: Extension from IR to a metric space with order or something like that. Secondly, the extension is not difficult and very elementary to prove.

(3) Is it interesting?

This is of course a highly subjective thing. If it somehow reads like a series of more or less simple facts and fairly obvious consequences, then it tends to be uninteresting. If a non-trivial concept is introduced and linked to known concepts, it tends to be more interesting. If an equation is treated where the case studied so far is quite obviously generalizable to the situation presented, it tends to be less interesting. If mathematics is simply done in an arbitrary way and generalized only for the sake of generalization, it is rather uninteresting. If a concrete problem is solved or a common structure for certain things is discovered, it is rather interesting. If it is claimed that the problem is interesting for applications, it must of course actually be so: An argument of the form “Addition is important for counting. We consider the following generalization of addition. This is important for counting.” is only okay if the generalization of addition is really important for counting. There is no borrowing of importance in this sense. Does the work describe a situation that occurs at all and differs significantly from known situations? It is possible to talk about non-Lebesgue measurable sets using the axiomatic system, but if they do not exist, then it is rather useless.

(4) Is the work sufficiently well written down?

This is also subjective, of course, but if you find a lot of mistakes, it's not good. Careless minor mistakes are okay (if there aren't too many of them), but major flaws in the argument are not. In that case, the paper is more likely to be rejected. For such a step, the mistake must of course be sufficiently substantial. (Not excluding the empty set is not such an error). Are the arguments transparent enough or do you have to know a lot of work in detail to understand them? (Phrases like “from the proof in [...] follows...” are not okay if they occur frequently).

(5) Conclusion

Finally, you write a review. This should contain the following things (this applies to works that are not already excluded due to other very, very substantial errors, in order to take a closer look at them; in this case, see below):

a) Title

Name of the paper, name of the authors. Important: Spelling must be correct here! (Later, typos are okay; but not in the names or the work)

b) Summary

A short description of the content of the work in your own words: What is it about? What is being done? What is the main result(s)? Maybe also: What techniques are used? The rather short MathSciNet summaries or the abstracts of the papers.

c) Analysis and Assessment

An analysis and personal assessment according to the four points above. It is sufficient to address the points you don't like. If there is need for praise, you can put this into the decision section below.  A list of found errors and misprints. Your critique of the work can also be formulated in several (enumerated) points, which is always good for resubmission because it allows the author to address the individual points specifically.

d) Decision.

There are basically four categories (regardless of the journal): i) Accept ii) Accept with minor Revision iii) Major Revision iv) Reject

In case of iii), you should insist on including the points of criticism raised and then you look at the work again with an open mind.

In case of ii), the editor usually does the final check.

If the work contains very gross mistakes that do not justify the above process, then you name the mistakes and reject the work after a short summary.

Again, that's a rough guideline for how to write reviews. It also helps to talk to other people about it and ask what their strategy is.