Linear Estimate for the Number of Zeros of Abelian Integrals
نویسندگان
چکیده
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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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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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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We consider the number of zeros of the integral I(h) = ∮ Γh ω of real polynomial form ω of degree not greater than n over a family of vanishing cycles on curves Γh : y 2 + 3x − x = 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[n−1 2 ] + 1.
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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 proble...
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ژورنال
عنوان ژورنال: Qualitative Theory of Dynamical Systems
سال: 2016
ISSN: 1575-5460,1662-3592
DOI: 10.1007/s12346-016-0213-0