Fuchs inequalities for systems of linear differential equations with regular singular points (Q941987)

Let \[ \frac{{dy}}{{dz}} = B(z)y\tag{1} \] be a system of \(p\) linear differential equations with the matrix \(B(z)\) whose entries are meromorphic on the Riemann sphere \(\bar C\) and holomorphic outside the set of singular points \(a_1 , \dots ,a_n.\) Suppose also that the matrix \(B(z)\) has the following Laurent expansion in a neighborhood of a singular point \(a_i \neq \infty\) : \[ B(z) = \frac{{B_{ - r_i - 1}^i }}{{(z - a_i )^{r_i + 1} }} + \cdots + \frac{{B_{ - 1}^i }}{{z - a_i }} + B_0^i + \cdots \] The author refines the Fuchs inequalities for system (1) obtained by \textit{E. Corel} [Inégalités de Fuchs pour les systèmes differéntiels réguliers, C. R. Acad. Sci. Paris, Sér. I,328, 983--986 (1999; Zbl 0983.34081)). Theorem. For the sum \(\Sigma\) of exponents of the system (1) with regular singular points \(a_1 , \ldots ,a_n \) of Poincaré rank \(r_1 , \dots ,r_n \), respectively there holds the inequality \[ -\frac{{p(p - 1)}}{2}\sum\limits_{i = 1}^n {r_i } + \sum\limits_{i = 1}^{n}{ \frac{{k_i (k_i - 1)}}{2}} \leq \Sigma \leq - \sum\limits_{i = 1}^n {\text{ rank}} B_{ - r_i - 1}^i r_i, \] where \(k_i = p - {\text{ rank}} B_{ - r_i - 1}^{i} = \dim \ker B_{ - r_i - 1}^{i} \) if \(r_i > 0\) and \(k_i = 0\) if \(r_i = 0\).