Approximation and complexity: Liouvillean-type theorems for linear differential equations on an interval
Let u, nu be solutions on an interval I of linear differential equations (LDEs) P = 0, Q = 0, respectively. We obtain a lower bound on the approximation of nu by u in terms of bounds on the coefficients of LDE S-i = 0 (for several i), satisfied by the ith derivative of nu and by the ith derivative of a basis of the LDE P = 0. One could view this result as a differential analog of the Liouville theorem which states that two different algebraic numbers are well separated if they satisfy algebraic equations with small enough integer coefficients. Unlike the algebraic situation, in the differential setting, in order to bound from below the difference \u - nu\, we need to involve not only the coefficients of P, Q themselves, but also those of S-i.
approximation and complexity liouvillean-type theorems linear differential equations