A counterexample for Barkatou's conjecture on the exponential growth order of solutions for Moser irreducible system and surgery operations (Algebraic analytic methods in complex partial differential equations)

Abstract

We, first, give a general result on the estimate of maximal exponential growth order $rho$(L) of solutions of Ly=0 with L=z^{p+1}(d/dz)I_{N}-A(z)(A(z)in M_{N}(mathrm{C}{z}) which is Moser irreducible defined by F. Moser [Mos]. Next, we give a counterexample for the conjecture by M. Barkatou on the characterization of $rho$(L) stated in his lecture [Bar]. We, further, introduce a class of system transformations called surgery operations by which the leading term of exponential factor of the formal fundamental matrix solution is calculated exactly for the obtained counterexample.