Towards the Proof of Yoshida's Conjecture

Abstract

Yoshida's Conjecture formulated by H. Yoshida in 1989 states that in $\mathbb{C}^{2N}$ equipped with the canonical symplectic form $\mathrm{d}\mathbf{p} \wedge \mathrm{d} \mathbf{q},$ the Hamiltonian flow corresponding to the Hamiltonian function \begin{equation*} H = \frac{1}{2}\sum_{i=1}^{N} p_{i}^{2} \sum_{i=1}^{N} p_{i}^{2} + \sum_{i=0}^{N} (q_i - q_{i+1})^k, \ \quad \text{with\ } q_0 = q_{N+1} = 0, \label{system} \end{equation*} where $N\geq 3$ is odd and $k\geq 4$ is even, has no meromorphic first integral functionally independent of $H$. For $N=3$ and $N=5$ with $k\geq 4$ arbitrary even number, the result was proved true by Maciejewski, Przybylska and Yoshida in 2012 by means of differential Galois theory. However, the question whether Yoshida's conjecture is true in general, remained open. In this paper we give a proof that this conjecture is in fact true for infinitely many values of $N$ using the results of R. D. Costin which are based on the so-called poly-Painlev\'{e} method devised by M. Kruskal.