The positivity problem for fourth order linear recurrence sequences is decidable

Abstract

The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.

A remark about the positivity problem of fourth order linear recurrence sequences

“…Proof. This is one of the hardest cases in our earlier work [11]. Let λ 3 = |λ 3 |e iθ , C = |C|e iϕ where θ, ϕ ∈ [−π, π), θ / ∈ {−π, 0} so that…”

“…If |C| = 0, then u n = Aλ n 1 + Bλ n 2 , which is of the form (HHH1) in [11,Lemma 2.3], and so is decidable.…”

“…However, certain partial results have already appeared, viz., the Positivity Problem for sequences satisfying a second order linear recurrence relation was shown to be decidable by Halava-Harju-Hirvensalo [5] in 2006. The Positivity Problem for sequences satisfying third and fourth order linear recurrences was shown to be decidable in [6], [11] and [12], respectively, see also [9,10]. As pointed out by Professor J. Ouaknine in a private communication, there is a gap in the proof of Claim 2 on page 141 of [11].…”

“…The Positivity Problem for sequences satisfying third and fourth order linear recurrences was shown to be decidable in [6], [11] and [12], respectively, see also [9,10]. As pointed out by Professor J. Ouaknine in a private communication, there is a gap in the proof of Claim 2 on page 141 of [11]. Our objective here is to repair this gap and a few loose arguments by giving a new proof, supplementing the works in [11,12], that the positivity problem for a fourth order linear recurrence sequence is decidable for one of the hardest cases where its characteristic polynomial has two distinct real and a complex conjugate pair of roots, and exactly one real root has the same absolute value as the two complex conjugate roots.…”

“…As pointed out by Professor J. Ouaknine in a private communication, there is a gap in the proof of Claim 2 on page 141 of [11]. Our objective here is to repair this gap and a few loose arguments by giving a new proof, supplementing the works in [11,12], that the positivity problem for a fourth order linear recurrence sequence is decidable for one of the hardest cases where its characteristic polynomial has two distinct real and a complex conjugate pair of roots, and exactly one real root has the same absolute value as the two complex conjugate roots. The proof, though different in some details, uses the same approach as in [10], i.e., invoking upon a deep result about linear forms in logarithms, and was suggested by Professor J. Ouaknine.…”