  We introduce an invariant of a finite point configuration $A \subset
\mathbb{R}^{1+n}$ which we denote the cuspidal form of $A$. We use this
invariant to extend Esterov's characterization of dual defective point
configurations to exponential sums; the dual variety associated to $A$ has
codimension at least $2$ if and only if $A$ does not contain any iterated
circuit.
