An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system

Document Type : Research Paper


1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 84156-83111

2 Department of Mathematical Sciences, University of Kashan, Kashan, Iran, 87317-53153


In this paper, we study the Chebyshev property of the 3-dimentional vector space $E =\langle I_0, I_1, I_2\rangle$, where $I_k(h)=\int_{H=h}x^ky\,dx$ and $H(x,y)=\frac{1}{2}y^2+\frac{1}{2}(e^{-2x}+1)-e^{-x}$ is a non-algebraic Hamiltonian function. Our main result asserts that $E$ is an extended complete Chebyshev space for $h\in(0,\frac{1}{2})$. To this end, we use the criterion and tools developed by Grau et al. in \cite{Grau} to investigate when a collection of Abelian integrals is Chebyshev.