On Liouvillian solutions of homogeneous linear differential equations

ABSTRACT

This paper deals with the problem of finding Liouvillian solutions of an nth order homogeneous linear differential equation L(y)=0 with coefficients in a differential field k whose field of constants is C. For second order linear differential equations such an algorithm has been given by J. Kovacic and implemented. A general decision procedure for finding Liouvillian solutions of nth order equations has been given by M.F. Singer, but the resulting algorithm, although constructive, is not in implementable form even for second order equations. The algorithm uses the fact that, if L(y)=0 has a Liouvillian solution, then, L(y)=0 has a solution z such that u=z′/z is algebraic over k, which means that L(y) has a solution z of the form e∫u, where u is algebraic over k. Since the logarithmic derivative u=z′/z of a solution z is a solution of the Riccati equation R(y)=0 associated to L(y)=0, the problem thus reduces to find an algebraic solution u of R(y)=0. This task is now split into two parts: to find the set DEG(n) of possible degrees N for the minimal polynomial P(x)=0 of u over k. to compute, for each possible degree of P(x), the possible coefficients of P(x). If we denote c(ii) the complexity of the second step and #DEG(n) the size of the set DEG(n), we see that the complexity of the whole procedure is of the form c(ii)#DEG(n) and thus exponential in #DEG(n). This shows that the only way to make the procedure effective is to get sharp bounds on the size of the set DEG(n), which is the scope of this paper. Initially, we construct, using representation theory of linear groups, a set DEG(n) where all N are of bounded size and only divisible by a small set of primes. Because of the divisibility condition the size of the set DEG(n) is asymptotically small. We derive the upper bound 2n4·&pgr;(n+1) for #DEG(n), where n is the degree of L(y)=0 and &pgr;(n) denotes the number of primes less or equal to n. This improves the bound of #DEG(n) from Jordan's theorem. The bound on the size of the primes that divide N is also a bound for the primes that divide the algebraic degree of the logarithmic derivative of at least one solution of L(y)=0. From the conditions on the size of the primes, some structure of the differential galois group can also be derived. Next, we study the action of the differential galois group on u to get sharp bounds for #DEG(n). The resulting set DEG(n) is the best possible one for n=2 and probably small enough to allow an implementation of the Singer-algorithm for n=3. We show that, for an algebraic solution u of R(u)=0, the degree N of the minimal polynomial P(x) of u equals the size of the orbit of u under the action of the differential galois group of L(y)=0. We then bound the size of the orbit of u by a value that can be effectively computed from a classification of the finite subgroups of PGL(n,C).

REFERENCES

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Nonalgebraic algorithms

General Terms:
Algorithms