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I read recently that there exists an Abel-Runge lemma. What is it? Google does not give an answer. I know about Abel's lemma (the summation by parts) and the Runge-Kutta method but I have never heard about Abel-Runge lemma before.

Update. This is a history of mathematics question. Apparently there was a result of Abel which was later modified (generalized?) by Runge. I do not know the work of Runge well enough to find out what it was. Hence the question.

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  • $\begingroup$ Could you give a link or reference to the Russian text? It would help to know what the lemma states, or at least is about. It might be better known under a different name. $\endgroup$
    – Conifold
    Sep 26 at 9:06
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    $\begingroup$ Is this the text? Judging by Google translation the author is just randomly putting some mathematical names into a playful verse, along with lots of other names. $\endgroup$
    – Conifold
    Sep 26 at 9:22
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    $\begingroup$ @Conifold: As a Russian speaker, I do not need Google translate, and confirm your conclusion: this is not a mathematical text, and the author just plays with names. $\endgroup$ Sep 26 at 11:56
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    $\begingroup$ "The translation from Russian is that there exist Dedekind calculations, an Abel-Runge lemma, Young modulus, etc. I am not sure it would help if I included that in the question." I think it might help to include a reference, as well as the relevant quotation and translation. Certainly, no harm can occur, at the very least. $\endgroup$ Sep 27 at 7:00
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    $\begingroup$ @TheAmplitwist: Thank you for suggestion. $\endgroup$
    – markvs
    Sep 27 at 7:12
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I found two papers by Runge about solving polynomial equations (one published in 1885, Runge, C. On the solvable equations of the form $x^5+ u x + v = 0$ (German) Acta Math. 7 (1885), no. 1, 173-186, and one published in 1963, about 40 years after Runge's death, by Ostrovski, Runge, Carl Eine Vorzeichenregel in der Theorie der algebraischen Gleichungen. (German) Jber. Deutsch. Math.-Verein. 66 (1963/64), Abt. 1, 52–66). None of them mentions Abel. So I guess the Abel-Runge lemma does not exist and this answers my question.

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    $\begingroup$ It is not unimaginable that, despite Runge never mentioning Abel, people started calling some result the Abel-Runge lemma at some point, so this does not completely settle the matter (I realize that settling it completely is next-to-impossible). $\endgroup$
    – Danu
    Sep 27 at 8:42
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    $\begingroup$ @Danu: What I wrote means that in these papers there are no statements which resemble anything Abel did (otherwise Runge would mention Abel). You can find these papers on the Web and see for yourself. $\endgroup$
    – markvs
    Sep 27 at 8:44
  • $\begingroup$ I also think that the lemma does not exist, but the conclusion here seems to be drawn on a thin basis. Abel did a lot more than just work on polynomial equations, e.g. series summation, complex analysis, elliptic functions. Runge also worked on complex analysis, among other things, and has a theorem there named after him. For a definitive conclusion the survey would have to be much more comprehensive. $\endgroup$
    – Conifold
    Sep 27 at 23:01
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    $\begingroup$ You may want to edit the text of the post to reflect that, along with an explanation why this is "the only area". $\endgroup$
    – Conifold
    Sep 28 at 0:50
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    $\begingroup$ This reminds me of the situation with the "Gauss-Manin connection". Gauss was not responsible for this concept (even in the slightest degree). The name was introduced by Grothendieck when generalizing a construction by Manin (from algebraic curves to higher-dimensional varieties). Why did Grothendieck decide to add Gauss' name to this notion remains a mystery (at least, to me). Maybe he simply wanted to name something after Gauss. :) $\endgroup$ Oct 1 at 18:58

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