APPLIED ALGEBRA An algorithm for computing invariants of differential Galois groups’

This paper presents an algorithm to compute invariants of the differential 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 integer m, one can compute the elements of the symmetric power Sym”‘ ( V(L)) that are left fixed by the Galois group. The bottleneck of previous methods is the construction of a differential operator called the ‘symmetric power of L’. Our strategy is to split the work into first a fast heuristic that produces a space that contains all invariants, and second a criterion to select all candidates that are really invariants. The heuristic is built by generalizing the notion of exponents. The checking criterion is ob-tained by converting candidate invariants to candidate dual first integrals; this conversion is done efficiently by using a symmetric power of a formal solution matrix and showing how one can reduce significantly the number of entries of this matrix that need to be evaluated. @ 199