Endlessly flat scalar potentials and $\alpha$-attractors

We consider a minimally-coupled inflationary theory with a general scalar potential $V(f(\varphi))= V(\xi\sum_{k=1}^{n}\lambda_k \varphi^k)$ containing a stationary point of maximal order $m$. We show that asymptotically flat potentials can be associated to stationary points of infinite order and discuss the relation of our approach to the theory of $\alpha$-attractors.