Deciding the positivity of a sequence defined by a linear recurrence and initial conditions is, in general, a hard problem. When the coefficients of the recurrences are constants, decidability has only been proven up to order 5. The difficulty arises when the characteristic polynomial of the recurrence has several roots of maximal modulus, called dominant roots of the recurrence. We study the positivity problem for recurrences with polynomial coefficients, focusing on sequences of Poincar\'e type, which are perturbations of constant-coefficient recurrences. The dominant eigenvalues of a recurrence in this class are the dominant roots of the associated constant-coefficient recurrence. Previously, we have proved the decidability of positivity for recurrences having a unique, simple, dominant eigenvalue, under a genericity assumption. The associated algorithm proves positivity by constructing a positive cone contracted by the recurrence operator. We extend this cone-based approach to a larger class of recurrences, where a contracted cone may no longer exist. The main idea is to construct a sequence of cones. Each cone in this sequence is mapped by the recurrence operator to the next. This construction can be applied to prove positivity by induction. For recurrences with several simple dominant eigenvalues, we provide a condition that ensures that these successive inclusions hold. Additionally, we demonstrate the applicability of our method through examples, including recurrences with a double dominant eigenvalue.

翻译：判定由线性递推及初始条件定义的序列的正性通常是一个困难问题。当递推系数为常数时，可判定性仅在阶数不超过5时得到证明。这一困难主要出现在递推的特征多项式具有多个模最大的根（称为递推的主导根）的情形。我们研究具有多项式系数的递推的正性问题，重点关注Poincaré型序列，即常系数递推的扰动形式。此类递推的主导特征值即为对应常系数递推的主导根。此前，我们在一般性假设下证明了具有唯一简单主导特征值的递推的正性可判定性，相关算法通过构造被递推算子压缩的正锥来证明正性。本文将这种基于锥的方法推广到更广泛的递推类别，其中压缩锥可能不再存在。核心思想是构造一个锥序列，使得递推算子将序列中的每个锥映射到下一个锥。该构造可通过归纳法用于证明正性。对于具有多个简单主导特征值的递推，我们给出了确保这种逐次包含关系成立的条件。此外，我们通过算例（包括具有双重主导特征值的递推）展示了该方法的适用性。