[Joint work with Antoine Colin]

Let be given a finite subgroup of a general linear group. We propose a
general frame to compute efficiently in the algebra of polynomials
invariant under this subgroup. The classical Noether normalization of
this Cohen-Macaulay algebra appears here naturally, and 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
Lagrange resolvents associated to the given subgroup. In particular we
show also how to find square-free resolvents with better theoretical
complexity.

Geometrically this relies on a geometric link between the discriminant
of the natural Noether projection and two other discriminants related
to fundamental invariants.