  In a model with more than one scalar doublet, the parameter space encloses
both physical and unphysical information. Invariant theory provides a detailed
description of the counting and characterization of the physical parameter
space. The Hilbert series for the 3HDM is computed for the first time using
partition analysis, in particular Omega calculus, giving rise to the
possibility of a full description of its physical parameters. A rigorous
counting of the physical parameters is given for the full class of models with
$N$ scalar doublets as well as a decomposition of the Lagrangian into
irreducible representations of $\mathrm{SU}(N)$. For the first time we derive a
basis-invariant technique for counting parameters in a Lagrangian with both
basis-invariant redundancies and global symmetries.
