Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

► Exact results for all stationary points of some high-dimensional function are given. ► They are interpreted as Gribov copies of a lattice Landau gauge fixing functional. ► The Gribov ambiguity and the Neuberger problem in compact U(1) are illustrated. ► Stationary points are used to discuss a criterion on the absence of phase transitions.