AN ALGORITHM FOR COMPUTING INVARIANTS OF DIFFERENTIAL GALOIS-GROUPS

This paper presents an algorithm to compute invariants of the differen tial Galois group of linear differential equations L(y)=0: if V(L) is the vector space of solutions of L(y)=0, we show how given some intege r m, one can compute the elements of the symmetric power Sym(m)(V(L)) that are left fixed by the Galois group. The bottleneck of previous me thods is the construction of a differential operator called the 'symme tric power of L'. Our strategy is to split the work into first a fast heuristic that produces a space that contains all invariants, and seco nd a criterion to select all candidates that are really invariants. Th e heuristic is built by generalizing the notion of exponents. The chec king criterion is obtained by converting candidate invariants to candi date dual first integrals; this conversion is done efficiently by usin g a symmetric power of a formal solution matrix and showing how one ca n reduce significantly the number of entries of this matrix that need to be evaluated. (C) 1997 Published by Elsevier Science B.V.