I introduce generally applicable strategies for the construction of basis invariants in multi-Higgs models. Applied to the two Higgs doublet model (2HDM), this allows us to find the full set of generating (CP-even and CP-odd) basis invariants. The complete ring of basis invariants is constructed, and novel relations between the CP-odd basis invariants emerge. This allows to derive the basis invariant necessary and sufficient conditions for explicit CP (non-)conservation in the 2HDM.
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