Non-interlaced solutions of 2-dimensional systems of linear ordinary differential equations

Abstract

We consider a 2-dimensional system of linear ordinary differential equations whose coefficients are definable in an o-minimal structure R. We prove that either every pair of solutions at 0 of the system is interlaced or the expansion of R by all solutions at 0 of the system is o-minimal. We also show that if the coefficients of the system have a Taylor development of sufficiently large finite order, then the question of which of the two cases holds can be effectively determined in terms of the coefficients of this Taylor development.