Vacuum Stability of a General Scalar Potential of a Few Fields

We find analytical vacuum stability or bounded below conditions for general scalar potentials of a few fields. After a brief review of copositivity we go beyond it. We discuss the vacuum stability conditions of the general potential of two real scalars, without and with the Higgs boson included in the potential. As further examples, we give explicit vacuum stability conditions for the two Higgs doublet model with no explicit CP breaking, and for the $\mathbb{Z}_{3}$ scalar dark matter with an inert doublet and a complex singlet. We give a short overview of positivity conditions for tensors of quartic couplings via tensor eigenvalues. A Mathematica notebook with the conditions is included with the source files.