On the multivariate Fujiwara bound for exponential sums

TL;DR

This paper establishes a multivariate Fujiwara bound for exponential sums, linking the amoeba's points to the tropical variety with explicit distance bounds depending on the sum's degree and scaling parameter.

Contribution

It extends the Fujiwara bound to multivariate exponential sums and provides explicit distance bounds between amoebas and tropical varieties.

Findings

Distance from amoeba points to tropical variety is bounded by a function of degree and scaling.

Improved bounds are provided for polynomial exponential sums.

The bounds depend explicitly on the sum's degree and the scaling parameter.

Abstract

We prove the multivariate Fujiwara bound for exponential sums: for a $d$-variate exponential sum $f$ with scaling parameter $\mu$, if $x$ is contained in the amoeba $\mathscr{A}(f)$, then the distance from $x$ to the Archimedean tropical variety associated to $f$ is at most $d \sqrt{d}\, 2\log(2 + \sqrt{3})/ \mu$. If $f$ is polynomial, then the bound can be improved to $d \log(2 + \sqrt{3})$.