Edit:For a recent progress on the Hilbert 16th problem see the following note which consider an infinite dimensional nature for this apparently 2 dimensional amazing problem . Best wishes for the authors of this paper and their final success in removing the mistake.(Can one guess that what is the mistake of the current(first) version of this paper?) I thank Loic Teyssier who informed me of existence of this paper.Any way, the first version of my question was the following:

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.

**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

entiremap from $\mathbb{C}$ to $\mathbb{C}P^{2}$? Can every leaf be parametrized by an entire map?

According to comments and answers to this question, there is no a written paper which explain the **error**, explicitly.**Why really this is the case?**