An algorithm for computing invariants of differential Galois groups Original Research Article

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 Symm(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 obtained 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.