×

On the reduction of the degree of linear differential operators. (English) Zbl 1232.34115

Authors’ abstract: Let \(L\) be a linear differential operator with coefficients in some differential field \(k\) of characteristic zero with an algebraically closed field of constants. Let \(k^a\) be the algebraic closure of \(k\). For a solution \(y_{0}\), \(Ly_{0} = 0\), we determine the linear differential operator of minimal degree \(\tilde L\) and coefficients in \(k^a\), such that \(\tilde Ly_0=0\). 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.

MSC:

34M03 Linear ordinary differential equations and systems in the complex domain
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv