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 Mat. Sb. (N.S.), 1970, Volume 83(125), Number 3(11), Pages 452–455 (Mi msb3521)

The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles

Yu. S. Ilyashenko

Abstract: Let $\mathrm A^R_n$ denote the coefficient space of the equations $\frac{dy}{dx}=\frac{P_n(x,y)}{Q_n(x,y)}$, $(x,y)\in R^2$, where $P_n$ and $Q_n$ are polynomials of degree $n\geqslant2$, and let $M_k$ denote the set of equations $\alpha\in\mathrm A^R_n$ that have limit cycles of multiplicity not less than $k$. For $2\leqslant k\leqslant\frac{n(n+1)}2$ the set $M_k$ is not empty. A proof is given for the
Theorem. The set $M_k$ does not form a semialgebraic manifold.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 12:3, 453–457

Bibliographic databases:

UDC: 517.9
MSC: 34C07, 14M20

Citation: Yu. S. Ilyashenko, “The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles”, Mat. Sb. (N.S.), 83(125):3(11) (1970), 452–455; Math. USSR-Sb., 12:3 (1970), 453–457

Citation in format AMSBIB
\Bibitem{Ily70} \by Yu.~S.~Ilyashenko \paper The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles \jour Mat. Sb. (N.S.) \yr 1970 \vol 83(125) \issue 3(11) \pages 452--455 \mathnet{http://mi.mathnet.ru/msb3521} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=276543} \zmath{https://zbmath.org/?q=an:0215.14702} \transl \jour Math. USSR-Sb. \yr 1970 \vol 12 \issue 3 \pages 453--457 \crossref{https://doi.org/10.1070/SM1970v012n03ABEH000930}