Geometric properties of the Riemann surfaces associated with the Noumi-Yamada systems with a large parameter
The Noumi-Yamada systems \((NY)_l\), \(l=2,3,4,\dots\), are systems of \(l+1\) first order ODEs for \(l+1\) unknowns which enjoy the affine Weyl group symmetry of type \(A_l^{(1)}\). These systems can be understood as the generalizations of the classical Painlevé equations PII, PIV and PV. Using appropriate scaling, it is possible to introduce the large parameter \(\eta\) and to study particular \(0\)-parameter solutions to \((NY)_l\) using the so-called exact WKB analysis. The above mentioned \(0\)-parameter formal series in \(\eta^{-1}\) is completely specified by the leading order term satisfying a system of algebraic equations. Earlier, the authors proved the existence of a \(0\)-parameter solution in the case of even \(l\). In the present paper, they treat the cases of even and odd \(l\) in a unified manner proving that the relevant algebraic system determines a variety which is isomorphic to the variety determined by a hyperbolic system with an appropriate normalization. In this way, the authors find the number of the possible leading order terms (i.e.\ the number of the branches of a multi-valued holomorphic function satisfying the algebraic system) outside the set of the turning points of the first kind (zeros of the Jacobian of the algebraic system) as well as the number of such turning points.
scientific article; zbMATH DE number 5992428