Unit roots of the unit root $L$-functions

TL;DR

This paper proves a conjecture that the unit root of certain $L$-functions associated with toric exponential sums behaves similarly to classical cases, extending previous results under a specific hypothesis.

Contribution

It generalizes the proof of Haessig and Sperber's conjecture on the behavior of the unit root $L$-function for toric exponential sums without the lower deformation hypothesis.

Findings

Confirmed the conjecture for all cases beyond the lower deformation hypothesis.

Extended the understanding of unit roots in $L$-functions related to toric exponential sums.

Linked the behavior of these unit roots to classical $L$-function cases.

Abstract

Adolphson and Sperber characterized the unique unit root of $L$-function associated with toric exponential sums in terms of the $\mathcal{A}$-hypergeometric functions. For the unit root $L$-function associated with a family of toric exponential sums, Haessig and Sperber conjectured its unit root behaves similarly to the classical case studied by Adolphson and Sperber. Under the assumption of a lower deformation hypothesis, Haessig and Sperber proved this conjecture. In this paper, we demonstrate that Haessig and Sperber's conjecture holds in general.