Abstract
The paper provides a sharpened proof of M. G. Khudai-Verenov 's theorem on the density in C2 of solutions to the equation dη/dξ=P/Q on condition that this equation has two singular points at infinity whose characteristic numbers satisfy certain constraints of the incommensurability type.
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M. G. Khudai-Verenov, “On one method of solving one differential equation,” Matem. Sb.,56, No. 3, 301–380 (1962).
I. G. Petrovskii and E. M. Landis, “On the number of limit cycles of the equation dy/dx = P(x, y) Q(x,y)−1, where P and Q are 2nd degree polynomials,” Matem. Sb.,37, No. 2, 209–250 (1955).
C. L. Siegel, “Iteration of analytic functions,” Ann. of Math., (2),43, No. 4, 607–612 (1942).
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Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 741–750, December, 1968.
In conclusion, we wish to thank our scientific co-worker E. M. Landis for many discussions of this work, as well as Ya. G. Sinai and M. L. Gerver,whose comments made it possible to improve the first part.
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Il'yashenko, Y.S. Density of an individual solution and ergodicity of a family of solutions to the equation dη/dξ=P(ξ, η)/Q(ξ, η). Mathematical Notes of the Academy of Sciences of the USSR 4, 934–938 (1968). https://doi.org/10.1007/BF01110832
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DOI: https://doi.org/10.1007/BF01110832