Optimal supports for rational invariants separating orbits of a C-torus
Résumé
I compute a minimal rational separating set of invariants for a representation of a torus (G× )d . That is, these rational invariants separate as many orbits as the invariant field. In particular, they separate more orbits than the invariant algebra.
To that purpose, I predefine a set of optimal supports for the invariants. The size of these optimal separating supports is bounded by d + 1. I thus obtain a separating set of cardinality bounded by \binom{n}{d+1}
∼ n^{d+1} , where n is the dimension of the representation. This is highly competitive when compared to other separating or generating sets, as exhibited in the final explicit examples.
The proofs rely on technical considerations of arithmetic and linear algebra, addressing the intersections of families of affine hyperplanes over Q that I call foliages.