Newton polygons for certain two variable exponential sums

TL;DR

This paper explicitly determines the Newton polygon for the L-function of a family of two-variable exponential sums, providing evidence for Wan's limit conjecture in a non-ordinary, multi-variable setting.

Contribution

It introduces a systematic method to compute the Newton polygon for a specific two-variable exponential sum family, extending previous work to non-ordinary cases.

Findings

Explicit Newton polygon obtained for the family

Provides asymptotic behavior supporting Wan's conjecture

Extends methods to non-ordinary multi-variable sums

Abstract

We studies the Newton polygon for the L-function of toric exponential sums attached to a family of two variable generalized hyperkloosterman sum,$f_{t}(x,y)=x^{n}+y+\frac{t}{xy}$ with $t$ the parameter. The explicit Newton polygon is obtained by systematically using Dwork's $\theta_{\infty}$-splitting function with an appropriate choice of basis for cohomology following the method of Adolphson and Sperber[2]. Our result provides a non-trivial explicit Newton polygon for a non-ordinary family of more than one variable with asymptotical behavior, which gives an evidence of Wan's limit conjecture[15].