Zeros of hyperelliptic integrals of the first kind for special hyperelliptic Hamiltonians of degree 7

In this paper, we study the Chebyshev's property of the three-dimensional vector space 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(x−47)(x−1)2 and (b) V′(x)=x(x−27)(x−1)4,E is an extended complete Chebyshev system. To this end, we use the criterion and the tools developed by Grau-Mañosas-Villadelprat in Trans. Amer. Math. Soc. in 2011.