On liouvillian solutions of linear differential equations

Abstract

This paper deals with the problem of finding liouvillian solutions of a homogeneous linear differential equationL(y)=0 of ordern with coefficients in a differential fieldk. For second order linear differential equations with coefficients ink o(x), wherek o is a finite algebraic extension ofQ, such an algorithm has been given by J. Kovacic and implemented. A general decision procedure for finding liouvillian solutions of a differential equation of ordern has been given by M.F. Singer, but the resulting algorithm, although constructive, is not in implementable form even for second order equations. Both algorithms use the fact that, ifL(y)=0 has a liouvillian solution, then,L(y)=0 has a solutionz such thatu=z′/z is algebraic overk. Using the action of the differential galois group onu and the theory of projective representation we get sharp bounds ℱ(n) for the algebraic degree ofu for differential equations of arbitrary ordern. For second order differential equations we get the bound ℱ(2)=12 used in the algorithm of J. Kovacic and for third order differential equation we improve the bound given by M.F. Singer from 360 to ℱ(3)≦36. We also show that not all values less than or equal to ℱ(n) are possible values for the algebraic degree ofu. For second order differential equations we rediscover the values 2, 4, 6, and 12 used in the Kovacic Algorithm and for third order differential equations we get the possibilities 3,4, 6, 7, 9, 12, 21, and 36. We prove that if the differential Galois group ofL(y)=0 is a primitive unimodular linear group, then all liouvillian solutions are algebraic. From this it follows that, if a third order differential equationL(y)=0 is not of Fuchsian type, then the logarithmic derivative of some liouvillian solution ofL(y)=0 is algebraic of degree 3. We also derive an upper bound for the minimal numberN(n) of possible degreesm of the minimal polynomial of an algebraic solution of the riccati equation associated withL(y)=0.