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 variants. 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 breakthroughs 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存在的开放性问题：线性递推序列能否从给定初始配置出发达到零？类似地，正性问题则询问线性递推序列从初始配置出发是否始终保持正值。判定Skolem问题（或正性问题）已悬置半个世纪之久：目前已知的可判定性结果仅适用于具有特殊性质（如低阶递推）的线性递推序列。但对于"未初始化"变体——即初始配置不固定而可任意变化的情形——这些问题会变得更容易：判断是否存在使线性递推序列保持正值的初始配置可在多项式时间内完成判定（Tiwari于2004年、Braverman于2006年证明）。本文研究介于初始化和未初始化变体之间的问题。具体而言，我们探讨对于给定初始配置的任意邻域初始配置，是否都能避免达到零（或避免出现负值）。这可视为Skolem问题（及正性问题）的鲁棒性变体。我们证明这些问题处于可判定性的边界：若邻域作为输入给出，则鲁棒Skolem问题和鲁棒正性问题均具有丢番图难度——即解决其中任一问题都需要在丢番图逼近领域取得重大突破，这与（非鲁棒）正性问题的情况类似。然而，若仅询问此类邻域是否存在，则问题可在PSPACE复杂度内判定。我们的技术方法还能处理最终正性问题的鲁棒性变体，该问题关注是否存在某个步数界限，使得在此界限之后线性递推序列始终保持正值。根据是否要求该步数界限具有"一致性"，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，该问题仍然是可高效求解的。