On the reduction of the degree of linear differential operators

Abstract

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let ka be the algebraic closure of k. For a solution y0, Ly0 = 0, we determine the linear differential operator of minimal degree and coefficients in ka, such that . This result is then applied to some Picard–Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka–Volterra type.