Monodromy problem related to Wu-Sutherland equations. (Q2751971)

Let \(\beta'\) be an \(n \times n\) matrix with real elements \(\beta_{i,j} '\) and \(z=(z_1,\dots,z_n)\) a point in \(\mathbb C^n\). Wu and Sutherland investigated the solutions \(w_i(z)\) of the equation (denoted WSe) NEWLINE\[NEWLINEw_i-1=z_i w_1^{\beta_{1i} '}\cdots w_n^{\beta_{ni}'}NEWLINE\]NEWLINE for \(1\leq i \leq n\). The author of the paper under review also investigates this equation, and, moreover, considers the countable family of equations NEWLINE\[NEWLINEw_i-1=e^{\sum_{j=1}^n(2\pi i) \beta_{ji} '\nu_j }z_i w_1^{\beta_{1i} '}\cdots w_n^{\beta_{ni}'}NEWLINE\]NEWLINE for \(1\leq i \leq n\) and given \(n\) integers \(\nu_1,\dots,\nu_n\). The purpose of the paper is to give a monodromy formula for the solutions of WSe. First the symmetry properties of WSe are investigated. The symmetries form a finite group of order \(6^n\cdot n!\) which is isomorphic to the semi-direct product of \((S_3)^n\) and \(S_n\). Under a certain nondegeneracy condition, the author gives \(3^n\) kinds of local solutions to WSe near the points \(w=(w_1,\dots,w_n)\) such that \(w_j=0, 1, \infty\) in \(\mathbb {CP}^n\). Their monodromy is described. Then an \(n\)-dimensional real variety \(X_{C_1,\dots,C_n}\) in the complex affine space \(\mathbb C^n\) is introduced and its properties are given. In conclusion the author states a conjectural theorem on the analytic continuation of germs of solutions of the two equations in a neighborhood of \((0,\dots,0)\) along a path in the variety \(X_{C_1,\dots,C_n}\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00054].