## Positivity problems for low-order linear recurrence sequences

*Joël
Ouaknine* and *James Worrell*
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.

*Proceedings of SODA 14*, 2014. 14 pages.

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