Norm-inflation for periodic NLS equations in negative Sobolev spaces

In this paper we consider Schr{"o}dinger equations with nonlinearities of odd order 2$\sigma$ + 1 on T^d. We prove that for $\sigma$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{"o}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3.