On sections of commutative group schemes
We prove that the semialgebraic, algebraic, and algebraic nonsingular points of a definable set in o-minimal structure with analytic cell decomposition are definable. Moreover, the operation of taking semialgebraic points is idempotent and the degree of complexity of semialgebraic points is bounded.
We prove that for algebras obtained by tilts from the path algebras of equioriented Dynkin diagrams of type Aₙ, the rings of semi-invariants are polynomial.