Twisted exponential sums of polynomials in one variable

TL;DR

This paper investigates the properties of twisted T-adic exponential sums for polynomials in one variable, establishing explicit bounds for their Newton polygons based on polynomial exponents, which inform bounds on related L-functions.

Contribution

It introduces an explicit arithmetic polygon as a lower bound for the Newton polygon of the associated C-function, linking polynomial exponents to exponential sum properties.

Findings

The arithmetic polygon provides a lower bound for the Newton polygon.

The bounds apply to the L-function of twisted p-power exponential sums.

Explicit relations between polynomial exponents and Newton polygons are established.

Abstract

The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the twisted T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of twisted $p$-power order exponential sums.