Decidability of positivity for C-finite sequences of order 6

Determine whether the positivity problem for C-finite sequences—i.e., sequences defined by linear recurrences with constant coefficients—of order 6 is decidable. Establishing decidability at this order would extend the known boundary beyond order 5 and clarify the connection to Diophantine approximation barriers.

Background

The paper reviews the status of the positivity problem for C-finite sequences (linear recurrences with constant coefficients). Solutions have the form of finite sums of polynomial–exponential terms determined by the roots of the characteristic polynomial. Despite this explicit structure, decidability of positivity is only known up to order 5.

The authors note that the order-6 case is tied to open problems in Diophantine approximation, indicating a barrier to current techniques. Resolving decidability at order 6 would mark a significant advance and could illuminate connections between positivity and Diophantine approximation.