Defective dual varieties for real spectra

TL;DR

This paper introduces the cuspidal form invariant for finite point configurations and extends Esterov's characterization of dual defectiveness from polynomial to exponential sums, linking geometric properties to combinatorial conditions.

Contribution

The paper defines a new invariant called the cuspidal form and extends dual defectiveness characterization to exponential sums, broadening geometric understanding.

Findings

Dual variety has codimension at least 2 iff no iterated circuit in A

Cuspidal form invariant characterizes dual defectiveness

Extension of Esterov's results to exponential sums

Abstract

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.