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What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?

Please see this related post

Added : According to their method, what of the following two statements are true?:

There are uniform numbers $\tilde{H}(n)$ such that every polynomial vector field $X$ of degree $n$ satisfies:

Statement 1) There are at most $\tilde{H}(n)$ real limit cycles of $X$ which lie on the same leaf.

statement 2) There are at most $\tilde{H}(n)$ distinct complex leaves which contains real limit cycles.

By "Leaf" I mean the leaf of the corresponding complex singular foliation of $\mathbb{C}P^{2}$. Some technical and historical aspects of these foliations are explained here.

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2 Answers 2

Not an answer, but way too long for a comment:

According to Ilyashenko ("Centennial history of Hilbert's 16th problem," http://www.ams.org/journals/bull/2002-39-03/S0273-0979-02-00946-1/S0273-0979-02-00946-1.pdf), the claimed result of Petrovski and Landis was disproved by Ilyashenko and Novikov (pg. 303). A citation to this disproof is not given, Ilyashenko's bibliography lists no articles by Ilyashenko & Novikov, and I can't find a paper by Ilyashenko and Novikov on the subject, so I don't believe they ever published their argument; however, Ilyashenko does point out that Landis and Petrovski claimed $H(2)=3$, but Shi Song Lin and Chen & Wang independently constructed quadratic vector fields with four limit cycles, showing that $H(2)\ge 4$. Their methods were based on Poincare Bendixon theorem on $S^{2}$ and the order of weak focus of quadratic system.

  • So what was the error?

This seems annoyingly hard to find! Landis and Petrovski's paper "On the number of limit cycles of the equation ${dy\over dx}={P(x, y)\over Q(x, y)}$, where $P$ and $Q$ are polynomials of 2nd degree" is freely available online (http://www.mathnet.ru/links/b7f25c4ee0acc2e5f39e5614ea6e4c54/sm5216.pdf); unfortunately, it seems to only be available in Russian, which I can't read. Additionally, I can't find a copy of Chen & Wang's paper or of Shi's paper online (although the abstract of the latter is available), which I would suspect say at least a little bit about Landis and Petrovski's arguments.

On the plus side, explicit constructions of quadratic vector fields with four limit cycles - even really nice ones! - are available in English: e.g., http://arxiv.org/pdf/1002.1055v1.pdf. I can't seem to find the simplest examples, though, which would presumably be the nicest for trying to figure out (without access to the paper itself) what the error was.


As a side question,

  • When was it noticed?

Ilyashenko says he and Novikov found the error "in the early 60s" (pg. 303). Backing this up, Shi's abstract mentions that the question "Is there a quadratic vector field with exactly 4 limit cycles?" was asked at a 1974 symposium on Hilbert's problems, so presumably the error was already known (Landis & Petrovski published in 1955). Then things get a bit tricky for me to track down: presumably, the symposium in question was "Mathematical developments arising from Hilbert's problems" (http://www.ams.org/bookstore-getitem/item=PSPUM-28); however, the table of contents for the proceedings of that symposium show no talk or paper on the 16th problem. Thus, although I'd imagine more specific discussion of when the error was noticed (and maybe even what it was!) would have happened at this symposium, I can't find it.

Beyond that, I have no idea when exactly the error was discovered.

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12  
It might be worth mentioning Landis, E. M.; Petrovskiĭ, I. G., A letter to the editors (Russian), Mat. Sb. (N.S.) 73 (115) 1967 160, MR0210979 (35 #1864), in which the authors acknowledge an error in the proof of Lemma 12 in their 1955 paper. The original Landis-Petrovsky paper in English translation in Petrovskiĭ, I. G.; Landis, E. M., On the number of limit cycles of the equation dy/dx=P(x,y)/Q(x,y), where P and Q are polynomials of the second degree, 1958 Amer. Math. Soc. Translations, Ser. 2, Vol. 10 pp. 177–221, American Mathematical Society, Providence, R.I., MR0094521 (20 #1036). –  Gerry Myerson Jun 17 at 4:38
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Another source that might be worth a look is Ilyashenko and Yakovenko, eds., Concerning the Hilbert 16th Problem, Translations AMS 2:165 (1995). –  Gerry Myerson Jun 17 at 4:39
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@GerryMyerson Thank you very much for your comment about the error in lemma 12 in their paper. –  Ali Taghavi Jun 17 at 14:26
    
@Noah Thanks for your very interesting answer. I remember that I read a small part of their english paper, befor my graduation. I remember an statement in the paper" if two limit cycles on the same leaf are homologe, then the leaf is an algebraic leaf and a generic algebraic equation does not have an algebraic solution..." –  Ali Taghavi Jun 17 at 14:32
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According to an interview of Ilyashenko, he found the flaw in the spring 1963. A few months later, Novikov was giving a series of talks on Petrovski-Landis' proof in Gelfand's seminar, and Ilyashenko told him about the mistake. Novikov later responded to some attempts by Landis to save the proof, and he was mentioned in the withdrawal letter as the one who found the flaw, "because Novikov was a renowned scholar and I was merely a grad student, so it would be awkward to put both names in the letter" –  Kostya_I Aug 10 at 17:21

Ilyashenko explaines very well the strategy and the main error of Petrovski-Landis in this lecture:

http://www.mathnet.ru/php/seminars.phtml?option_lang=rus&presentid=8316

Watch from 50m.

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1  
thank you very much for the video. There is a problem in my download system. Can I save it in my computer after downloading? Did you watched the part of Petrovski Landis strategy?can you sketch this strategy and ITS Main ERROR? Do you know what is the true version of their strategy?My deep thanks to you for the video. –  Ali Taghavi Jul 12 at 0:20

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