Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

Abstract

We propose a new method to compute the unipotent radical $R_u(H)$ of the differential Galois group $H$ associated to a parameterized second-order homogeneous linear differential equation of the form \[\tfrac{\partial^2}{\partial x^2}Y-qY=0,\] where $q \in F(x)$ is a rational function in $x$ with coefficients in a $Π$-field $F$ of characteristic zero, and $Π$ is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of $R_u(H)$ to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quotient $H / R_u(H)$ is a $Π$-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations $Π'$ such that the associated differential Galois group $H'$ has the property that $H'/ R_u(H')$ is $Π'$-constant, and such that $R_u(H)$ is defined by the same differential equations as $R_u(H')$. Thus the computation of $R_u(H)$ is reduced to the effective computation of $R_u(H')$. We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.