Oscillation of Linear Ordinary Differential Equations: on a Theorem by A. Grigoriev

We give a simplified proof and an improvement of a recent theorem by A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions to systems of linear equations with polynomial or rational coefficients. 1. Background on counting zeros of solutions of linear ordinary differential equations 1.1. De la Valée Poussin theorem and Novikov’s counterexample. A linear nth order homogeneous differential equation y + a1(t) y (n−1) + · · ·+ an−1(t) y + an(t) y = 0, y = d y dtk , (1.1) with real continuous coefficients aj(t) is called disconjugate (Chebyshev, nonoscillating) on a real interval [α, β] ⊆ R, if any nontrivial solution y(t) of this equation has at most n− 1 isolated root on this interval. A theorem by C. de la Vallée Poussin [dlVP29] asserts that any equation (1.1) is disconjugate on any interval sufficiently short relative to the magnitude of the coefficients of the equation. More precisely, if |aj(t)| 6 bj on [α, β] and ∑n j=1 bj |β−α|j/j! < 1, then (1.1) is disconjugate. This allows to place a rather accurate upper bound on the number of isolated roots of any solution of a known differential equation in terms of the length of the interval on which the solution is considered and the magnitude of the coefficients of the equation. A complex analog of this theorem was obtained in [Yak99] for linear homogeneous nth order equations with holomorphic coefficients in a polygonal complex domain t ∈ U ⋐ C. Date: September 2004. 2000 Mathematics Subject Classification. 34C08, 34C10, 34M10.