On the Chebyshev Property of a Class of Hyperelliptic Abelian Integrals: On the Chebyshev Property of a Class of Hyperelliptic

• Published in: Qualitative theory of dynamical systems 2024, Vol.23 (Suppl 1)
• Main Authors: Sun, Yangjian, Wang, Shaoqing, Yang, Jiazhong
• Format: Article
• Language: English
• Subjects: Difference and Functional Equations; Dynamical Systems and Ergodic Theory; Mathematics; Mathematics and Statistics
• ISSN: 1575-5460; 1662-3592
• DOI: 10.1007/s12346-024-01136-3

This paper aims to demonstrate the Chebyshev property of the linear space V = { ∑ i = 0 2 α i ∮ Γ h x 2 i y d x : α 0 , α 1 , α 2 ∈ R , h ∈ Σ } (which is equivalent to that every function of V has at most 2 zeros, counted with multiplicity), with three hyperelliptic Abelian integrals ∮ Γ h x 2 i y d x ( i = 0 , 1 , 2 ) as generators, where Γ h is an oval determined by H ( x , y ) = y 2 2 + Ψ ( x ) = h , and Ψ ( x ) is an even polynomial of indefinite degree with real non-Morse critical points. As an application, we can obtain the exact upper bound for the number of zeros of a class of hyperelliptic Abelian integrals related to some planar polynomial Hamiltonian systems with two cusps and a nilpotent center.