It is "known" that Écalle and Ilyiashenko (independently) solved the (first part of the) second part of Hilbert's 16th problem (i.e. they proved that the number of limit cycles of a polynomial vector field in the real plane is finite). However: apart from their respective books, is there any other more accessible proof?
I am an expert on the topic and I fear that both books are not read very much...