The error in Petrovski and Landis' proof of the 16th Hilbert problem 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  and also the following post.. For  Mathematical  development around this  historical problem please see this  paper.
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. However in this linked paper there is no an explicit explianation about the "error".
According to the video of lecture of Ilyashenko, provided in the answer to this question  by Andrey Gogolev, we ask: 

What is the fate of the "persistence problem" which is mentioned by Ilyashenko? How it can be revised to become a true statement?

According to the first page of the english version of the paper of Petrovski_Landis we ask 

"How they assume that a solution of the equation can be  considered  as  an  entire map  from $\mathbb{C}$  to $\mathbb{C}P^{2}$?  Can every leaf be parametrized by  an entire  map? Does this "entire-assumption" play an important role in their proof? Please See this  related post.

According to comments and answers  to this question, we undrestand there is  no  a  written paper which explains the error, explicitly.Why really  this is the case?
 A: • Q1 (the first yellow boxed question in the OP): 
The current status of the "persistence problem" has been discussed by Ilyashenko in Complex length and persistence of limit cycles in a neighborhood of a hyperbolic polycycle (2014), see also Persistence Theorems and Simultaneous Uniformization (2006).   
As discussed on page 285 of the 2014 paper, the persistence theorem introduced by Petrovskiĭ and Landis (P&L), without a valid proof, asks whether the real limit cycle of a real planar polynomial vector field persists as a complex limit cycle for the complexification of the corresponding differential equation. Ilyashenko notes that the theorem is still unproven for the general family of polynomial equations, but does give a proof for a particular case (and references to other special cases).    
In general, the persistence theorem applies if a complex singular foliation of ${\mathbb C}{\mathbb P}^2$ satisfies a certain condition of "simultaneous uniformization", explained in the 2006 paper. The error in Petrovskiĭ & Landis amounts to the absence of a proof that polynomial differential equations in the plane are simultaneously uniformizable.
 The 2006 paper also goes some way towards an answer of the request in the OP for "a written paper which explains the error in P&L".
In connection with this recent MO question asking for Examples of incorrect arguments being fertilizer for good mathematics?, the P&L paper is a good example, as also pointed out here.
• historical note: according to V.Gaiko, the error in Petrovskiĭ and Landis's proof was first noted by the Belorussian mathematician N.P. Erugin.
A: 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.
A: 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.
