Tropical Varieties for Exponential Sums

TL;DR

This paper investigates the geometric and computational complexity of approximating zero sets of exponential sums, providing explicit bounds and complexity results, including undecidability and polynomial-time algorithms for approximations.

Contribution

It introduces a polyhedral approximation for zero sets of exponential sums and analyzes its accuracy and computational complexity, including bounds and decidability results.

Findings

The real part of zero sets can be approximated by a polyhedral skeleton.

An explicit Hausdorff distance bound is established.

Membership for the zero set is undecidable in general, but polynomial-time for the approximation.

Abstract

We study the complexity of approximating complex zero sets of certain $n$-variate exponential sums. We show that the real part, $R$, of such a zero set can be approximated by the $(n-1)$-dimensional skeleton, $T$, of a polyhedral subdivision of $\mathbb{R}^n$. In particular, we give an explicit upper bound on the Hausdorff distance: $\Delta(R,T) =O\left(t^{3.5}/\delta\right)$, where $t$ and $\delta$ are respectively the number of terms and the minimal spacing of the frequencies of $g$. On the side of computational complexity, we show that even the $n=2$ case of the membership problem for $R$ is undecidable in the Blum-Shub-Smale model over $\mathbb{R}$, whereas membership and distance queries for our polyhedral approximation $T$ can be decided in polynomial-time for any fixed $n$.