Generic A-family of exponential sums

TL;DR

This paper constructs a generating polynomial that encodes the generic Newton polygon for a family of exponential sums, providing a tool to understand their properties across large primes.

Contribution

It introduces a generating polynomial over the rationals for the generic Newton polygon of exponential sums of the family f = x^d + a x^s, revealing key properties.

Findings

The generating polynomial encodes the generic Newton polygon for large primes.

It determines the Newton polygon at each prime p when p is sufficiently large.

The polynomial reveals key properties of the exponential sums' L functions.

Abstract

In this paper we construct a generating polynomial over the rationals for the generic Newton polygon for the L function of exponential sums of the family of f = x^d+ a x^s parameterized by a, and prove some of its key properties. The generating polynomial encodes information of and determines the generic Newton polygon at each prime p when p is large enough, and vice versa.