Fuchs' relation for irregular differential systems (Q2748286)

The Fuchs relation for a linear differential of order \(n\) with coefficients in \(\mathbb{C}(z)\) with regular singularities on the Riemann sphere is a classical formula which imposes a constraint to the sum of the exponents of the singular points. So, the sum of the exponents should be \(1/2(m-1)n(n-1)\), where \(m+1\) is the number of singular points including the infinity [\textit{E. G. C. Poole}, Introduction to the theory of linear differential equations, Oxford: Clarendon Press (1936; Zbl 0014.05801 and JFM 62.1277.01), p. 77]. Several generalizations and extensions have been obtained recently. For instance, systems of linear differential equations with regular singular points satisfy a Fuchs relation, but now it is given by an inequality [\textit{A. A. Bolibrukh}, The 21st Hilbert problem for linear differential equations, Proc. Steklov Inst. Math 206 (1995; Zbl 0844.34004) and the author, Bull. Soc. Math. Fr. 129, 189--210 (2001; Zbl 0998.34075)].NEWLINENEWLINEIn the paper under review, a Fuchs inequality formula is obtained for systems of linear differential equations with irregular singular points and with coefficients in \(\mathbb{C}(z)\). The formula is given in the language of connections.