This paper deals with the polynomial linear system solving with errors (PLSwE) problem. Specifically, we focus on the evaluation-interpolation technique for solving polynomial linear systems and we assume that errors can occur in the evaluation step. In this framework, the number of evaluations needed to recover the solution of the linear system is crucial since it affects the number of computations. It depends on the parameters of the linear system (degrees, size) and on a bound on the number of errors. Our work is part of a series of papers about PLSwE aiming to reduce this number of evaluations. We proved in [Guerrini et al., Proc. ISIT'19] that if errors are randomly distributed, the bound of the number of evaluations can be lowered for large error rate. In this paper, following the approach of [Kaltofen et al., Proc. ISSAC'17], we improve the results of [Guerrini et al., Proc. ISIT'19] in two directions. First, we propose a new bound of the number of evaluations, lowering the dependency on the parameters of the linear system, based on work of [Cabay, Proc. SYMSAC'71]. Second, we introduce an early termination strategy in order to handle the unnecessary increase of the number of evaluations due to overestimation of the parameters of the system and on the bound on the number of errors.

中文翻译：

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具有随机误差的多项式线性系统求解：新界限和提前终止技术

本文涉及多项式线性系统的带误差求解（PLSwE）问题。具体而言，我们专注于求解多项式线性系统的评估插值技术，并假设在评估步骤中可能会发生错误。在此框架中，恢复线性系统解所需的评估数量至关重要，因为它会影响计算数量。它取决于线性系统的参数（度，大小），并且取决于误差的数量。我们的工作是有关PLSwE的一系列论文的一部分，旨在减少这种评估的数量。我们在[Guerrini et al。，Proc。ISIT'19]认为，如果错误是随机分布的，则对于较大的错误率，可以降低评估数量的范围。在本文中，遵循[Kaltofen等人，Proc.Natl.Acad.Sci.USA 90：5873-5877。ISSAC'17]，我们改善了[Guerrini et al。，Proc。ISIT'19]向两个方向发展。首先，我们根据[Cabay，Proc.Natl.Acad.Sci.USA，90：3877-3886]的工作，提出了评估数量的新界限，降低了对线性系统参数的依赖性。SYMSAC'71]。其次，我们引入了一种提前终止策略，以处理由于对系统参数的过高估计以及错误数量的界限而导致的不必要数量的增加。