The Chebyshev’s property of certain hyperelliptic integrals of the first kind

In this work, we study the Chebyshev’s property of the 3-dimensional vector space E=,E=, where Ji(h)=∫H=hxidxy and H(x,y)=12y2+V(x) is a hyperelliptic Hamiltonian of degree 7. Our main result asserts that in two specific cases, namely (a) V′(x)=x3(1−x)3V′(x)=x3(1−x)3 and (b) V′(x)=x5(x−1),V′(x)=x5(x−1),E is an extended complete Chebyshev space. To this end we use the criterion and the tools developed by Grau et al. in [6]. We pose also the conjecture that E   is also a Chebyshev space when V′(x)=x(x−1)5V′(x)=x(x−1)5. In this regard we give a partial result, Theorem 1.4, concerning the Chebyshev property of two subspaces of E. To prove it we use another criterion by Mañosas and Villadelprat [7] to study when a collection of Abelian integrals is Chebyshev with accuracy k.