The monotonicity of the ratio of two Abelian integrals

Liu, Changjian; Xiao, Dongmei

doi:10.1090/S0002-9947-2013-05934-X

The monotonicity of the ratio of two Abelian integrals
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by Changjian Liu and Dongmei Xiao PDF

Trans. Amer. Math. Soc. 365 (2013), 5525-5544 Request permission

Abstract:

In this paper, we study the monotonicity of the ratio of two Abelian integrals \[ I_0(h)=\int _{\Gamma _h}y dx\quad \textrm {and}\quad I_1(h)=\int _{\Gamma _h}xy dx,\] where $\Gamma _h$ is a compact component of the level set $\{(x,y):\ y^2+\Psi (x)=h, \ h\in J\}$; here $J$ is an open interval. We first give a new criterion for determining the monotonicity of the ratio of the above two Abelian integrals. Then using this new criterion, we obtain some new Hamiltonian functions $H(x,y)$ so that the ratio of the associated two Abelian integrals is monotone. Especially when $H(x,y)$ has the form $y^2+P_5(x)$, we obtain the sufficient and necessary conditions that the ratio of two Abelian integrals is monotone, where $P_5(x)$ is a polynomial of $x$ with degree five.

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