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This is community wiki question.

I will be writing my first review for ZBMath. I would like to take some suggestion through examples.

In general, abstract is too small and introduction is too lengthy in a paper to decide whether to read the paper or not.

I am under the impression that an ideal review for a (published) article should be more like an invitation from the referee to the user to read the paper, or, much better to be an appreciation letter.

Question : Are there any reviews in ZBMath (Mathscinet has restricted access) that motivated you to read the article. Can you give some examples.

I am aware of the question How to write a good MathSciNet review?. That discusses theoretical aspect of a good review but it not all of the answers have examples showing a good review.

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    $\begingroup$ I’m not sure I agree with the premise. I think ZBMath or MathSciNet reviews should aim to give a factual and neutral account of the results of the paper in question. Perhaps together with some background/context that might be missing from something as short as an abstract. $\endgroup$ Commented Dec 4, 2023 at 15:53
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    $\begingroup$ @SamHopkins I respect your point of view.. For reasons I can't explain quickly, I prefer to give only positive feedback in official public domain.. I was thinking anything I write should be helping the authors to get one more reader :) Is that not the usual stand of reviewers? I am not saying if it is good or not.. Just wanted to know what Is the usual stand.. $\endgroup$ Commented Dec 4, 2023 at 16:14
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    $\begingroup$ I agree with Sam Hopkins. Reviews are written for their readers, to give the readers accurate information about the papers, and not to promote authors. Even the authors would not be helped if readers find less in their papers than the reviews seemed to promise. $\endgroup$ Commented Dec 4, 2023 at 16:21
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    $\begingroup$ @PraphullaKoushik MathSciNet and zbMATH "reviews" are not like book reviews that you might read in a newspaper or a magazine, which readers typically read in order to learn the opinion of the reviewer. A zbMATH review is certainly not an endorsement either. Now, if you genuinely love a paper, then you should feel free to write an enthusiastic review. But the primary purpose of the review is to provide factual information about the paper. Think of it as an extended abstract which gives additional context, details, and search terms that the paper's own abstract lacked space to include. $\endgroup$ Commented Dec 5, 2023 at 13:10
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    $\begingroup$ My favorite reviews are those that make me feel I don't have to read the paper. An ideal review should describe key results, put them in proper context, and sketch the proofs. Reading the actual paper is for the few experts who care. $\endgroup$ Commented Dec 5, 2023 at 23:27

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It has been emphasized in the comments that a zbMATH or MathSciNet review is not an endorsement, and that unlike a "review" that one might find in a newspaper or a magazine, its primary purpose is not to present the reviewer's subjective opinion of the quality of the paper. Nevertheless, if you genuinely admire a paper, then you are not prohibited from providing subjective remarks.

One review (from MathSciNet rather than zbMATH) that I like to tell people about is Hans Volkmer's review of Marek Rychlik's paper, A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weitzenböck (Invent. Math. 129 (1997), 141–212). Volkmer's review begins as follows.

The well-known equichordal problem from geometry asks whether there exists an equichordal curve in the plane. The problem was posed by Fujiwara in 1916 and, independently, by Blaschke, Rothe and Weitzenböck in 1917. Here are all the definitions needed to understand the problem: (1) An equichordal point P of a Jordan curve (a simply closed curve) is a point inside the curve such that all chords (line segments) connecting two points of the curve and passing through P have the same length. (2) An equichordal curve is a Jordan curve that has two distinct equichordal points. The definition of an equichordal curve originally required it to be convex but this property was dropped later.

Since the equichordal problem is so easy to formulate and so accessible even to non-mathematicians, a large number of problem solvers tried their luck with it. But most of them quickly found out that the problem cannot be solved by just drawing a picture of an "equichordal curve'' and arriving at a contradiction by means of elementary geometry. Some "solutions'' were obtained in this way, though. The problem can even claim to be of practical importance because the reviewer knows an engineer who asked for a formula of an equichordal curve in order to develop a better Wankel engine. It was quite clear for some time that equichordal curves do not exist, but a proof was missing. At first sight it may appear strange that many people spent so much effort and time on showing that something does not exist. But on the other hand, work on challenging problems has often led to exciting new developments in mathematics.

As you can see, the reviewer does an excellent job of providing context for the paper. The remark about the Wankel engine is an intriguing fact that is probably not available from any other source, and piques the reader's interest.

The review then goes on to describe prior work, and to sketch the sophisticated method of proof. I'll skip this part of the review and just cut to the concluding lines:

As one would expect, the paper is long (72 pages) and involved. The reading is easy at the beginning but becomes more difficult towards the end of the paper.

The paper represents an extraordinary effort of the author that, without doubt, will find high appreciation by everyone in the mathematical world.

The final line is of course a subjective endorsement, but what makes this an excellent review is not the final line; it is rather the engaging way in which the context of the paper is described, as well as the careful choice of which details to present (some of which I have omitted here).

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    $\begingroup$ Many thanks for the detailed explanation, with example... :) $\endgroup$ Commented Dec 5, 2023 at 16:32

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