What is the status of the "infinitesimal Hilbert's sixteenth problem" (aka "Hilbert-Arnold Problem")? According to the Russian Wikipedia article, it was still open in 2009. I am interested in this because I'm attending an elementary course in dynamical systems for undergraduates, and I think I can more or less get a feel for the simplified statement of the problem now. Just out of curiosity, is it still an open problem in mathematics? If so, what are the main difficulties in solving it (why is it so hard)?
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2$\begingroup$ Related: mathoverflow.net/questions/310718/… $\endgroup$– David Roberts ♦Commented Sep 29 at 13:34
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A 2023 paper on the infinitesimal (weak) version of the 16th Hilbert problem is Infinitesimal and tangential 16-th Hilbert problem on zero-cycles. It states that the problem is still unsolved in the original formulation (vector fields in the plane), although a zero-dimensional version can be solved.
answered Sep 29 at 18:45