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 Tr. Mat. Inst. Steklova, 2006, Volume 254, Pages 215–246 (Mi tm110)

Weak Infinitesimal Hilbert's 16th Problem

I. A. Khovanskaya (Pushkar')

State University – Higher School of Economics

Abstract: The following weak infinitesimal Hilbert's 16th problem is solved. Given a real polynomial $H$ in two variables, denote by $M(H,m)$ the maximal number possessing the following property: for any generic set $\{\gamma _i\}$ of at most $M(H,m)$ compact connected components of the level lines $H=c_i$ of the polynomial $H$, there exists a form $\omega =P dx+Q dy$ with polynomials $P$ and $Q$ of degrees no greater than $m$ such that the integral $\int _{H=c}\omega$ has nonmultiple zeros on the connected components $\{\gamma _i\}$. An upper bound for the number $M(H,m)$ in terms of the degree $n$ of the polynomial $H$ is found; this estimate is sharp for almost every polynomial $H$ of degree $n$. A multidimensional version of this result is proved. The relation between the weak infinitesimal Hilbert's 16th problem and the following question is discussed: How many limit cycles can a polynomial vector field of degree $n$ have if it is close to a Hamiltonian vector field?

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English version:
Proceedings of the Steklov Institute of Mathematics, 2006, 254, 201–230

Bibliographic databases:

UDC: 517.927.7
Received in July 2005

Citation: I. A. Khovanskaya (Pushkar'), “Weak Infinitesimal Hilbert's 16th Problem”, Nonlinear analytic differential equations, Collected papers, Tr. Mat. Inst. Steklova, 254, Nauka, MAIK «Nauka/Inteperiodika», M., 2006, 215–246; Proc. Steklov Inst. Math., 254 (2006), 201–230

Citation in format AMSBIB
\Bibitem{Kho06} \by I.~A.~Khovanskaya (Pushkar') \paper Weak Infinitesimal Hilbert's 16th~Problem \inbook Nonlinear analytic differential equations \bookinfo Collected papers \serial Tr. Mat. Inst. Steklova \yr 2006 \vol 254 \pages 215--246 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm110} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2301007} \elib{http://elibrary.ru/item.asp?id=13517725} \transl \jour Proc. Steklov Inst. Math. \yr 2006 \vol 254 \pages 201--230 \crossref{https://doi.org/10.1134/S0081543806030102} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33749410349}