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未解决的开放问题：从给定初始配置出发，线性递归序列（LRS）能否达到0？类似地，正性问题询问LRS能否从初始配置始终保持正值。判定Skolem问题（或正性问题）已开放半个世纪：目前已知的可判定性结果仅针对具有特殊性质（如低阶递归）的LRS。但对于"非初始化"变体而言，这些问题更为简单：初始配置并非固定而是可以任意变化时，检查是否存在某个初始配置使LRS保持正值可在多项式时间内判定（Tiwari, 2004；Braverman, 2006）。本文研究介于初始化与非初始化变体之间的问题。具体而言，我们关注能否在给定初始配置的邻域内，对每个初始配置都避免0（或负数）。这可视作Skolem问题（或正性问题）的鲁棒变体。研究表明这些问题处于可判定性边界：若邻域作为输入给定，则鲁棒Skolem与鲁棒正性问题具有丢番图困难性，即解决任一问题都需要丢番图逼近领域的重大突破（如同非鲁棒正性问题）。然而若仅询问此类邻域是否存在，则问题可判定且复杂度为PSPACE。我们的技术还可处理终极正性的鲁棒性问题，即判断是否存在步数上界使LRS在此之后保持正值。根据该步数上界是否为"统一"值，存在两种变体。对于非统一变体，当邻域为开集时，即使邻域作为输入给定，问题仍可处理。