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 Mat. Sb. (N.S.), 1975, Volume 98(140), Number 3(11), Pages 363–377 (Mi msb3715)

On the density of solutions of an equation in $\mathbf{CP}^2$

B. Müller

Abstract: In this paper we consider the system
$$\dot u=P(u),$$
where $u=(u_0,u_1,u_2)\in\mathbf C^3$, $P=(P_0,P_1,P_2)$ and the $P_i$ are homogeneous polynomials of degree $2n$ ($n\geqslant1$) with complex coefficients. Let $A_n$ be the space of coefficients of the right-hand sides of the system (1). Any point $\alpha\in A_n$ defines a system of the form (1).
Our aim in this paper is to show that the property of the solutions of the system (1) being dense in $\mathbf{CP}^2$ is locally characteristic, i.e. we prove that in $A_n$ there exists an open set $U$ such that the solutions of the system (1) with right-hand side $\alpha\in U$ are everywhere dense in $\mathbf{CP}^2$.
This result can be extended without difficulty to the case in which the degree of the homogeneous polynomials appearing in the right-hand side of the system (1) is odd.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 27:3, 325–338

Bibliographic databases:

UDC: 517.92
MSC: Primary 34C05; Secondary 34A20

Citation: B. Müller, “On the density of solutions of an equation in $\mathbf{CP}^2$”, Mat. Sb. (N.S.), 98(140):3(11) (1975), 363–377; Math. USSR-Sb., 27:3 (1975), 325–338

Citation in format AMSBIB
\Bibitem{Mul75} \by B.~M\"uller \paper On~the density of solutions of an equation in~$\mathbf{CP}^2$ \jour Mat. Sb. (N.S.) \yr 1975 \vol 98(140) \issue 3(11) \pages 363--377 \mathnet{http://mi.mathnet.ru/msb3715} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=466689} \zmath{https://zbmath.org/?q=an:0319.34006} \transl \jour Math. USSR-Sb. \yr 1975 \vol 27 \issue 3 \pages 325--338 \crossref{https://doi.org/10.1070/SM1975v027n03ABEH002517} 

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This publication is cited in the following articles:
1. Ilyashenko Y., “Centennial History of Hilbert's 16th Problem”, Bull. Amer. Math. Soc., 39:3 (2002), 301–354
2. A. A. Shcherbakov, “Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on $\mathbb C^2$”, Proc. Steklov Inst. Math., 254 (2006), 103–120
3. J.E.rik Fornæss, Nessim Sibony, “Riemann Surface Laminations with Singularities”, J Geom Anal, 18:2 (2008), 400
4. Nataliya Goncharuk, Yury Kudryashov, “Genera of non-algebraic leaves of polynomial foliations of $\mathbb C^2$”, Mosc. Math. J., 18:1 (2018), 63–83
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