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Completing the picture for the Skolem Problem on order-4 linear recurrence sequences

Piotr Bacik

TL;DR

This work settles the Skolem problem for all algebraic order-4 linear recurrence sequences by combining decomposition into non-degenerate subsequences with a Kronecker-based analysis of dominant roots. It leverages Baker-type lower bounds on linear forms in logarithms to control zeroes and shows that four-root cases must lie in the MSTV class, where decidability is already established. The result closes the long-standing gap for order-4 LRS and solidifies the boundary between decidable and open cases in the Skolem problem. The methods integrate number-theoretic height theory, p-adic and Archimedean linear forms, and structural root analysis to achieve an effective decision procedure.

Abstract

For almost a century, the decidability of the Skolem Problem - that is, the problem of finding whether a given linear recurrence sequence (LRS) has a zero term - has remained open. A breakthrough in the 1980s established that the Skolem Problem is indeed decidable for algebraic LRS of order at most 3, and real algebraic LRS of order at most 4. However, for general algebraic LRS of order 4 the question of decidability has remained open. Our main contribution in this paper is to prove decidability for this last case, i.e. we show that the Skolem Problem is decidable for all algebraic LRS of order at most 4.

Completing the picture for the Skolem Problem on order-4 linear recurrence sequences

TL;DR

This work settles the Skolem problem for all algebraic order-4 linear recurrence sequences by combining decomposition into non-degenerate subsequences with a Kronecker-based analysis of dominant roots. It leverages Baker-type lower bounds on linear forms in logarithms to control zeroes and shows that four-root cases must lie in the MSTV class, where decidability is already established. The result closes the long-standing gap for order-4 LRS and solidifies the boundary between decidable and open cases in the Skolem problem. The methods integrate number-theoretic height theory, p-adic and Archimedean linear forms, and structural root analysis to achieve an effective decision procedure.

Abstract

For almost a century, the decidability of the Skolem Problem - that is, the problem of finding whether a given linear recurrence sequence (LRS) has a zero term - has remained open. A breakthrough in the 1980s established that the Skolem Problem is indeed decidable for algebraic LRS of order at most 3, and real algebraic LRS of order at most 4. However, for general algebraic LRS of order 4 the question of decidability has remained open. Our main contribution in this paper is to prove decidability for this last case, i.e. we show that the Skolem Problem is decidable for all algebraic LRS of order at most 4.
Paper Structure (8 sections, 10 theorems, 13 equations)

This paper contains 8 sections, 10 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic numbers and heights
  4. Linear recurrence sequences
  5. Linear forms in logarithms and the MSTV class
  6. Linear forms in logarithms
  7. The MSTV class
  8. The Skolem Problem at order 4

Key Result

Theorem 2.1

Any fractional ideal $I$ of $K$ has a unique decomposition where $t \in \mathbb{N}$, each $\mathfrak{p}_i \subseteq \mathcal{O}_K$ is a prime ideal and $n_i \in \mathbb{Z}$.

Theorems & Definitions (12)

  • Theorem 2.1: Unique Factorisation Theorem
  • proposition 2.2
  • Theorem 3.1: Matveev
  • Theorem 3.2: Yu
  • Theorem 3.3
  • Theorem 3.4
  • definition 3.5: MSTV class
  • Theorem 3.6
  • Theorem 4.1
  • Theorem 4.2: Kronecker
  • ...and 2 more