Zeros of complete elliptic integrals and its application to Melnikov functions

In this paper, we first discuss the linear independence of the complete elliptic integrals of the first, second and third kinds $K(k)$, $E(k)$ and $\mathrm{\Pi }(\mu (k),k)$, and then obtain an upper bound for the number of zeros of a function of the form \begin{eqnarray*} p(k)K(k)+q(k)E(k)+r(k)Π(μ(k),k),\ k\in(-1,1), \end{eqnarray*} where $p(k)$, $q(k)$ and $r(k)$ are real polynomials, $\mu (k)$ is a real polynomial or rational function. Finally, we apply it to a Hamiltonian triangle with three invariant straight lines under small real polynomials piecewise smooth perturbation.