Skolem Problem for Linear Recurrence Sequences with 4 Dominant Roots (after Mignotte, Shorey, Tijdeman, Vereshchagin and Bacik)

TL;DR

The work analyzes the Strong Skolem problem for non-degenerate linear recurrence sequences (LRS) of order four, focusing on cases with up to four distinct roots and using the dominant-root framework. It provides an expository synthesis of the analytic-number-theory machinery—absolute values, heights, Baker-type lower bounds via Matveev and Yu, and Beukers-type geometric tricks—to obtain effective lower bounds that force nonvanishing of the sequence beyond finitely many indices. The main steps decompose U(n) into dominant and tail terms and show that, under suitable nondegeneracy conditions, large n cannot yield U(n)=0; Bacik’s lemma further ensures a global dominance pattern enabling decidability for the order-4 case. Together, these results recap and consolidate the MST84Ve85 arguments (and Bacik’s refinement), establishing decidability of the Strong Skolem problem for non-degenerate LRS with at most four distinct roots, and thus for order up to four in many important circumstances.

Abstract

It is an exposition of the work of Mignotte, Shorey, Tijdeman, Vereshchagin and Bacik on decidability of the vanishing problem for linear recurrence sequences of order 4.

Figures (1)

• Figure 1: two circles

Theorems & Definitions (32)

• Proposition 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Proposition 2.1: Kronecker Theorem
• Remark 2.2
• Proposition 2.3
• Remark 2.4
• Corollary 2.5
• Proposition 2.6