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，线性递归序列（LRS）能否达到0？类似地，正性问题询问LRS是否在给定初始配置下保持正值。判定Skolem问题（或正性问题）已开放半个世纪：已知的可判定性结果仅适用于具有特殊性质的LRS（例如低阶递归）。但对于“未初始化”变体，这些问题更容易处理——即初始配置不固定，可任意变化：检查是否存在某个初始配置使得LRS保持正值可在多项式时间内判定（Tiwari, 2004；Braverman, 2006）。本文考虑介于初始化和未初始化变体之间的问题。更精确地，我们询问：在给定初始配置的邻域内，是否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。我们表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和鲁棒正性问题为丢番图困难问题——求解其中任何一个都将对丢番图逼近理论产生重大突破，这与（非鲁棒）正性问题的情形类似。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。我们的技术还允许处理最终正性的鲁棒性，该问题询问是否存在步数上界使得LRS在此之后保持正值。根据是否要求该步数的“一致”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，问题仍可被有效处理。