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What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?

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    $\begingroup$ (from wikipedia) the second part is: The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree $n$ and an investigation of their relative positions $\endgroup$ – YCor Sep 16 '18 at 21:00
  • $\begingroup$ @Ycor Thank you for adding the tag "limit cycle"! $\endgroup$ – Ali Taghavi Sep 16 '18 at 21:00
  • $\begingroup$ @YCor and also adding the wikipedia link and editting the title. $\endgroup$ – Ali Taghavi Sep 16 '18 at 21:01
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    $\begingroup$ Shall we play update the title to optimize the English? “Updated Background on Hilbert’s 16th Problem”? $\endgroup$ – Anthony Quas Sep 16 '18 at 23:09
  • $\begingroup$ @AnthonyQuas Thanks for your comment. Do you have a suggestion for revision of the title? $\endgroup$ – Ali Taghavi Sep 17 '18 at 20:55
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An update from April 2018 is given by Patrick Speissegger.

The idea, going back to Poincaré, is to reduce the two-dimensional counting problem (counting limit cycles in the plane) to a one-dimensional counting problem (counting certain points on a line). Roussarie (1998) showed that Hilbert’s 16th problem follows if a certain "finite cyclicity conjecture" holds. A tameness condition called "o-minimality" allows to reformulate Roussarie's conjecture as a conjecture of o-minimality. Speissegger discusses special cases where o-minimality can be proven and proposes this approach as a promising way to prove Hilbert's 16th problem.

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  • $\begingroup$ Dear Prof. Beenakker Thank you very much and my previous +1 for your valuable answer and sorryf or my late acceptance $\endgroup$ – Ali Taghavi Oct 26 '18 at 22:56

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