Tropical Varieties for Exponential Sums and their Distance to Amoebae
Alperen A. Ergür, Grigoris Paouris, J. Maurice Rojas · arXiv (Cornell University) · 2014
Given any $n$-variate exponential sum, $g$, the real part of the complex zero set of $g$ forms a sub-analytic variety $\Re(Z(g))$ generalizing the amoeba of a complex polynomial. We extend the notion of Archimedean tropical hypersurface to derive a piecewise linear approximation, $\mathrm{Trop}(g)$, of $\Re(Z(g))$, with explicit bounds --- solely as a function of $n$, the number of terms, and the minimal distance between frequencies --- for the Hausdorff distance $\Delta(\Re(Z(g)),\mathrm{Trop}(g))$. We also discuss the membership complexity of $\mathrm{Trop}(g)$ relative to the Blum-Shub-Smale computational model over $\mathbb{R}$. Along the way, we also estimate the number of roots of univariate exponential sums in axis-parallel rectangles, refining earlier work of Wilder and Voorhoeve.