Power residue symbols and the exponential local-global principle

TL;DR

This paper investigates the exponential local-global principle, also known as Skolem's conjecture, focusing on specific quadratic and degenerate cubic cases, and proves new results using power residue symbols.

Contribution

It introduces new proofs for quadratic and degenerate cubic cases of the conjecture utilizing power residue symbols, advancing understanding of the open problem.

Findings

Proved new quadratic cases of the exponential local-global principle.

Established degenerate cubic cases using power residue symbols.

Contributed to the partial resolution of Skolem's conjecture.

Abstract

The exponential local-global principle, or Skolem conjecture, says: Suppose that \(b\) is a positive integer, and that the sequence \((u_{n})_{n = -\infty}^{\infty}\) is such that every term is in \(\mathbb{Z}[1/b]\), the linear recurrence \(u_{n + d} = a_{1}u_{n + d - 1} + \cdots + a_{d}u_{n}\) holds for all integers \(n\), and every root of \(x^{d} - a_{1}x^{d - 1} - a_{2}x^{d - 2} - \cdots - a_{d}\) is nonzero and simple; then there is no zero term \(u_{n}\) if and only if, for some integer \(m\) that is larger than \(1\) and relatively prime to \(b\), every term \(u_{n}\) is not in \(m\mathbb{Z}[1/b]\). Particular cases of the conjecture are known, but the general conjecture is open. This paper proves some apparently new quadratic and degenerate cubic cases of the exponential local-global principle via power residue symbols. This work was presented at the Stellenbosch Number Theory Conference 2025 in January 2025 at Stellenbosch University; much of the work was also presented at the 67th Annual Congress of the South African Mathematical Society in December 2024 at the University of Pretoria.