# Moscow Mathematical Journal

Volume 20, Issue 2, April–June 2020  pp. 311–321.

Tropical Approximation of Exponential Sums and the Multivariate Fujiwara Bound

Authors:  Jens Forsgård (1)
Author institution:(1) Department of Mathematics, Texas A&M University, College Station, TX 77843

Summary:

We prove a multivariate analogue of the Fujiwara bound: for a $d$-variate exponential sum $f$ with exponents having spacing $\mu$, the distance from a point $x$ in the amoeba $\mathscr{A}_f$ to the Archimedean tropical variety of $f$ is at most $d \sqrt{d}\, 2\log(2 + \sqrt{3})/ \mu$. If all exponents are integral, then the bound can be improved to $d \log(2 + \sqrt{3})$. Both bounds are within a constant factor of optimal.

2010 Math. Subj. Class. Primary: 11L03; Secondary: 14T03.

Keywords: Fujiwara bound, exponential sum, amoeba, tropical variety.