High scale boundary conditions with an additional complex singlet

We investigate Planck scale boundary conditions on Higgs quartic interactions and their $\beta$-functions in the SM augmented by an additional complex scalar. We use renormalisation group running at two-loops, and include both theoretical and experimental constraints. We find that the boundary condition $\lambda=\beta_\lambda=0$ at the Planck scale is compatible with the current Higgs and top mass measurements, but requires additional scalars lighter than about $600\,$GeV.