  We consider semilinear Schr\"odinger equations with nonlinearity that is a
polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$
or on the torus. Norm inflation (ill-posedness) of the associated initial value
problem is proved in Sobolev spaces of negative indices. To this end, we apply
the argument of Iwabuchi and Ogawa (2012), who treated quadratic
nonlinearities. This method can be applied whether the spatial domain is
non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant
or not.
