The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for ``uninitialized'' variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variant. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthrough in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a ``uniform'' bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

翻译：Skolem问题是线性动力系统中一个长期悬而未决的开放问题：给定初始配置，线性递推序列（LRS）能否达到0？类似地，正性问题询问该序列是否从初始配置起始终保持正值。判定Skolem问题（或正性问题）已开放半个世纪：已知的最优可判定性结果仅适用于具有特殊性质（如低阶递推）的LRS。但这些问题对于"未初始化"变体较为容易，其中初始配置不是固定的，而是可以任意变化：判断是否存在一个初始配置使LRS始终保持正值可在多项式时间内判定（Tiwari, 2004；Braverman, 2006）。在本文中，我们考虑介于初始化和未初始化变体之间的问题。更精确地说，我们询问在给定初始配置的邻域内，是否从每个初始配置都能避免0（或负数）。这可以视为Skolem问题（或正性问题）的鲁棒变体。我们证明这些问题处于可判定性的边界：若邻域作为输入的一部分给出，则鲁棒Skolem问题和鲁棒正性问题均为丢番图困难问题，即解决其中任何一个都将导致丢番图逼近领域的重大突破，正如（非鲁棒）正性问题的情况。然而，若仅询问是否存在这样的邻域，则这些问题变得可判定且具有PSPACE复杂度。我们的技术还允许处理最终正性问题的鲁棒性，该问题询问是否存在一个步数上界，使得LRS在此后始终保持正值。根据是否要求该步数的"一致"上界，存在两种变体。对于非一致变体，当邻域为开集时，即使邻域作为输入给出，该问题也属于易解范畴。