We present the results of our recent article [4] and discuss its applications [5]. A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the compact torus. We study the structure of the orbit space as the image of the fundamental invariants. For the Weyl groups associated to crystallographic root systems of types A, B, C, D, this image is a compact basic semi-algebraic set. We give the defining polynomial inequalities explicitly as the positivity-locus of a Hermite matrix polynomial. As an application, we consider the problem of computing the optimal value of an exponential function and solve it with algebraic methods under symmetry assumptions.