Linear Estimate of the Number of Zeros of Abelian Integrals for a Kind of Quartic Hamiltonians*
نویسندگان
چکیده
where f (x, y) and g(x, y) are real polynomials of x and y with degree not greater than n, 1h is an oval lying on real algebraic curve H(x, y)=h, deg H(x, y)=m (H(x, y) are called Hamiltonians), and 7 is a maximal interval of existence of 1h . This question is called the weakened Hilbert 16th problem, posed by V.I. Arnold in [1, 2]. The general result of solving the weakened Hilbert 16th problem was achieved by A.Varchenko [12] and A. Khovanskii [6], who proved independently the existence of Z(m, n), but no explicit expression of Z(m, n) has been obtained. Yu. Ilyashenko, S. Yakovenko and D. Novikov have proved in [4, 5, 7, 13] that for the set of ``good'' H(x, y) there exists a Article ID jdeq.1998.3581, available online at http: www.idealibrary.com on
منابع مشابه
LINEAR ESTIMATE OF THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A KIND OF QUINTIC HAMILTONIANS
We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.
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