Monday, June 29, 2009

Re-request/when to include a known proof in a paper?

In my post of June 24 I requested some help on a sum (among other things). In particular, we had a summation result that we were using in a paper and we wanted to know if it was new or not. We got three comments relevant to it
1. Method of differences due to Pascal
2. discrete calculus analog of the fact that the nth degree Taylor expansion of a polynomial is the polynomials.
3. The first lemma is a direct consequence of the method of finite differences; it says that the nth forward difference of a polynomial of degree n-1 is identically zero. The inner summation in the second identity is basically computing the coefficients of the Newton Form of the polynomial, although you seem to be using a slightly different basis. Nevertheless, these results are quite standard.
This raises a request and some question.

REQUEST:
Authors of those comments (and others who know stuff)--- please email me more exact references. I found some things on Google but it would be good to know what to read and what is a good source to reference. Preferable online.
QUESTION: In a paper if you are using a result that is known how much detail should you put in?
1. Clearly put in that it is known and provide a references. I would not want to frustrate my readers with This is easily derived from the method of differences. without providing a reference. In this day and age an online reference if you can mange it.
2. If the result is not quite written down anywhere but your readers could easily derive it using known techniques, then you do not need to supply a proof. But this is not as clear a statement as it sounds- it depends on how you define readers, easily, known, and technique.
3. If the result is known but you have a cute proof of it which seems new (hard to tell) then what do you do?
4. If the proof is short then I am more inclined to include it. If its an e-journal than length matters less. (This topic has been raised in a different form before---if Conferences proceedings are CD's then why have a page limit?)
5. If there are no online references then I am more inclined to include a proof.
6. My only real point here is that its a QUESTION- what is the cutoff for what is worth including? There is no one answer.

1. Never hurts to include a new or even an old proof with the disclaimer "for completeness".

2. if there is a clean reference refer to it.

3. Always there is an option to create a longer version, with all the "well-known" details, on arXiv and give a link.

4. Always there is an option to create a longer version, with all the "well-known" details, on arXiv and give a link.

This is suboptimal, since it risks confusion between the two versions of the paper. (For example, someone may refer to the short version and think it contains material found only in the long version.) Instead of a longer version of the same paper, it should be a supplementary paper.

As for references, Concrete Mathematics has a lot of information about this. It's not online (at least not legally), but it's definitely a better reference for most purposes than anything available online on this subject.

Never hurts to include a new or even an old proof with the disclaimer "for completeness".

Occasionally it hurts, if the new proof is more contrived or complicated than the standard proofs, since that can give readers the wrong impression.

5. For a known result that you can find a proof somewhere in the literature, I think the decision should be similar to when to include a proof in a survey paper: yes if reading the proof teaches the reader something more than "this lemma is true", no if it doesn't. If it seems to be folklore but you can't track down a solid reference, though, probably best just to prove it with some disclaimer that you're not claiming any novelty for it.

6. This is part of what appendices are for.

7. I don't know of a reference (besides some material I've written myself), but the proofs I know are short enough that they should probably just be given.

Here's one way to think about the result: if p(x) is a polynomial of degree n, the first result is equivalent to the statement that \sum p(k) x^k is a rational function with denominator (1 - x)^{n+1} and the second result can (I believe) be obtained by computing the coefficients in the partial fraction decomposition thereof.

8. there is no justification to publish something that was already published.

9. I want to propose the opposite viewpoint to the majority, that it frequently hurts to include simple proofs for completeness. "Concrete" statements from mathematics like this are, as Serge Lang used to say, "trivial or false". Simple sums, integrals, etc..., are things that the reader can usually check fairly easily. On the other hand, including these results makes it harder to read the paper because you have to dig through all the concrete results to find where real advances are. Further, errors with papers are almost never in these simple concrete results; on the contrary, one main cause of mistaken papers is to spend too much time giving overly detailed steps of the little results without really checking over the new ideas.

10. On the other hand, including these results makes it harder to read the paper because you have to dig through all the concrete results to find where real advances are.

I think this is avoidable. Usually when I'm in such a situation, I put all these 'concrete results' in a section labeled "Preliminary Lemmas" or "Appendix" to draw the reader's attention away from them. So, the reader knows what's important, and for the curious reader who wants to understand all details doesn't see the 'trivial' proofs of the lemmas, they can flip to the appropriate section and read them.

11. Further, errors with papers are almost never in these simple concrete results; on the contrary, one main cause of mistaken papers is to spend too much time giving overly detailed steps of the little results without really checking over the new ideas.

I disagree. Students or amateurs often fall into error by spending too much time on trivialities and skimming over the new ideas, but for professionals it's usually the opposite.

In any case, the moral is clear: whatever you are tempted not to write up is the most likely place for an error to be hiding.