Efficient computation of square-free Lagrange resolvents

We propose a general frame to compute efficiently in the invariant algebra k[X1, . . . , Xn] , whereH is a finite subgroup of the general linear groupGLn(k). The classical Noether normalization of this Cohen-Macaulay algebra takes a natural form when expressed with adequate data structures, based on evaluation rather than writing. This allows to compute more efficiently its multiplication tensor. As an illustration we give a fast symbolic algorithm to compute the coefficients of the Lagrange resolvent associated to the given subgroup H, either generically or specialized. We show also how to find square-free resolvents with better theoretical complexity (polynomial in the index of the group after a precomputation depending only on H). This relies on a geometric link between the discriminant of the natural Noether projection and two other discriminants related to fundamental invariants.