Effective Positivity Problems for Simple Linear Recurrence Sequences

We consider two computational problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (determine whether all terms of a given LRS are positive) and the effective Ultimate Positivity Problem (determine whether all but finitely many terms of a given LRS are positive, and if so, compute an index threshold beyond which all terms 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, and effective Ultimate Positivity is solvable in polynomial time.