Smooth and Algebraic Invariants of a Group Action: Local and Global Constructions

We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan’s normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants. 1 Smooth construction Algebraic construction Consider Lie grp. G y M smooth manifold over R Alg. grp. G y Z ⊂ Kn affine variety smooth, local, semi-regular action rational action F(M)G-smooth invariants K(Z)G – rational invariants Construct local cross-section to the orbits graph-section ⇓ ⇓ moving frame map ρ :M → G reduced Gröbner basis ⇓ ⇓ fundamental set for F(M)G finite generating set for K(Z)G normalized inv. with replacement property replacement inv. that are K(Z)G-tuples ⇓ ⇓ projection ι : F(M)→ F(M)G projection ι : K(Z)→ K(Z)G