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On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
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