Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to~5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.

翻译：判定由多项式系数线性递推及初始条件定义的序列的正定性通常较为困难。即使在常系数递推情形下，已知可判定性仅局限于阶数不超过5的情况。本文研究一大类具有多项式系数的任意阶线性递推，针对初始条件位于超平面之外的情形，我们提出可判定正定性的算法。该算法通过构造一个被递推算子收缩的锥体，使得归纳法证明正定性成为可能。此类锥体的存在性与构造依赖于将经典Perron-Frobenius理论推广至保持锥体不变的矩阵理论。