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 = < J(0), J(1), J(2) >, where J(i)(h) = integral(H=h) X-i dx/y and H(x, y) = 1/2 y(2) + V(x) is a hyperelliptic Hamiltonian of degree 7. Our main result asserts that in two specific cases, namely (a) V' (x) = x(3) (1 - x)(3) and (b) V"(x) = x(5) (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 F, is also a Chebyshev space when V' (x) = x(x - 1)(5). In this regard we give a partial result, Theorem 1.4, concerning the Chebyshev property of two subspaces of F. To prove it we use another criterion by Manosas and Villadelprat [7] to study when a collection of Abelian integrals is Chebyshev with accuracy k. (C) 2015 Elsevier Ltd. All rights reserved.