On Kloosterman sums over finite fields of characteristic 3

TL;DR

This paper investigates the divisibility properties of Kloosterman sums over finite fields of characteristic 3, introducing a new recursive algorithm to determine the highest power of 3 dividing these sums, which simplifies zero detection.

Contribution

It presents a novel recursive algorithm for computing the maximal power of 3 dividing Kloosterman sums over characteristic 3 fields, enabling easier zero testing.

Findings

New recursive algorithm for divisibility by 3^k

Simplified zero detection for Kloosterman sums

Enhanced understanding of sum divisibility properties

Abstract

We study the divisibility by 3^k of Kloosterman sums K(a) over finite fields of characteristic 3. We give a new recurrent algorithm for finding the largest k, such that 3^k divides the Kloosterman sum K(a). This gives a new simple test for zeros of such Kloosterman sums.