The monotonicity of the ratio of two Abelian integrals

Abstract

In this paper, we study the monotonicity of the ratio of two Abelian integralswhere Γ h is a compact component of the level set {(x, y) : y 2 + Ψ(x) = h, h ∈ 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.

Parameter identification on Abelian integrals to achieve Chebyshev property

“…Claim 1: S 1 (x, α) has no zeros for x ∈ (0, 1) if α ∈ (α * , +∞), where α * is isolated in 2835 2048 , 5671 4096 and given in the proof. Claim 2: S 3 (x, α), which is given in (18), has two zeros counting multiplicity…”

“…A direct application of Lemma 2.3 and the symbolic computation analysis can only reach the upper bound to be 5, when α ∈ (−∞, α 1 ) ∪ (α * , +∞). The combination introduced leads to a comprehensive analysis on the ratio of the two Wronskians and its derivative, which includes the major factor S 3 (x, α) given in (18). It is surprising to find that S 3 (x, α) can be factorized as…”

“…The main tool used in this paper is an algebraic method to bound the number of zeros of linear combination of a set of integrals on a family of closed orbits, see [8,19]. It is a general version of the method of determining the monotonicity of two Abelian integrals [16,18]. A brief introduction is given below.…”

The cyclicity of a class of quadratic reversible centers defining elliptic curves

“…To address this question, Chebyshev criteria, an effective algebraic tool that sufficient conditions are established to determine an upper bound of the number of zeros of the Abelian integral, will be applied in this paper. As far as we know, Chebyshev criteria have been carried out in many classic studies, see [29,20,9,27,28] for example. Li and Zhang in [20] first defined criterion functions from the integrands of Abelian integral…”